Open Access

Diffraction problems for quasilinear parabolic systems with boundary intersecting interfaces

Boundary Value Problems20132013:99

DOI: 10.1186/1687-2770-2013-99

Received: 19 January 2013

Accepted: 9 April 2013

Published: 22 April 2013

Abstract

In this paper, we discuss the n-dimensional diffraction problem for weakly coupled quasilinear parabolic system on a bounded domain Ω, where the interfaces Γ k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq1_HTML.gif ( k = 1 , , K 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq2_HTML.gif) 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

1 Introduction

Let Ω be a bounded domain in R n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq3_HTML.gif with boundary Ω ( n 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq4_HTML.gif), and let Ω be partitioned into a finite number of subdomains Ω k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq5_HTML.gif ( k = 1 , , K https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq6_HTML.gif) separated by Γ k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq7_HTML.gif, where Γ k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq7_HTML.gif, k = 1 , , K 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq2_HTML.gif, are interfaces, which do not intersect with each other. For any T > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq8_HTML.gif, set
Q T : = Ω × ( 0 , T ] , S T : = Ω × [ 0 , T ] , Γ : = k = 1 K 1 Γ k , Γ T : = Γ × [ 0 , T ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_Equa_HTML.gif
In this paper, we consider the diffraction problem for quasilinear parabolic reaction-diffusion system in the form
{ u t l L l ( u l ) = g l ( x , t , u ) ( ( x , t ) Q T ) , [ u l ] Γ T = 0 , [ a i j l ( x , t , u l ) u x j l ν i ( x ) ] Γ T = 0 , u l = ψ l ( x , t ) ( ( x , t ) S T { Ω × { 0 } } ) , l = 1 , , N , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_Equ1_HTML.gif
(1.1)
where x = ( x 1 , , x n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq9_HTML.gif, u = ( u 1 , , u N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq10_HTML.gif, u t l : = u l / t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq11_HTML.gif, u x i l : = u l / x i https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq12_HTML.gif, u x l : = ( u x 1 l , , u x n l ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq13_HTML.gif,
L l ( u l ) : = d d x i ( a i j l ( x , t , u l ) u x j l ) + b j l ( x , t , u l ) u x j l , l = 1 , , N , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_Equ2_HTML.gif
(1.2)

repeated indices i or j indicate summation from 1 to n, ν ( x ) : = ( ν 1 ( x ) , , ν n ( x ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq14_HTML.gif is the unit normal vector to Γ (the positive direction of ν ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq15_HTML.gif is fixed in advance), the symbol [ ] Γ T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq16_HTML.gif denotes the jump of a quantity across Γ T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq17_HTML.gif, and the coefficients a i j l ( x , t , u l ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq18_HTML.gif, b j l ( x , t , u l ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq19_HTML.gif and g l ( x , t , u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq20_HTML.gif are allowed to be discontinuous on Γ T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq17_HTML.gif. In the following, we refer to the conditions on Γ T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq17_HTML.gif 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 [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 Γ k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq7_HTML.gif do not intersect with Ω, the existence and uniqueness of the solutions have been investigated in [1114] 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 [1114] 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, S ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq21_HTML.gif denotes its closure. The symbol Ω Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq22_HTML.gif means that Ω Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq23_HTML.gif and dist ( Ω , Ω ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq24_HTML.gif.

Let
{ Γ 1 , , Γ K 1 } = { Γ 1 , , Γ K 0 1 } { Γ 1 , , Γ K K 0 } , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_Equb_HTML.gif
where Γ k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq25_HTML.gif, k = 1 , , K 0 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq26_HTML.gif, intersect with the outer boundary Ω, and Γ k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq27_HTML.gif, k = 1 , , K K 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq28_HTML.gif do not intersect with Ω. Assume that the domain Ω is partitioned into subdomains Ω k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq29_HTML.gif, k = 1 , , K 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq30_HTML.gif, separated by interfaces Γ k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq25_HTML.gif, and partitioned into Ω k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq31_HTML.gif, k = 1 , , K K 0 + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq32_HTML.gif, separated by Γ k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq27_HTML.gif. The interface of Ω k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq29_HTML.gif and Ω k + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq33_HTML.gif is Γ k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq25_HTML.gif. Then Ω ¯ = k = 1 K 0 Ω ¯ k = k = 1 K K 0 + 1 Ω ¯ k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq34_HTML.gif. Set
Q k , T : = Ω k × ( 0 , T ] for  k = 1 , , K , Γ : = k = 1 K 0 1 Γ k , Γ : = k = 1 K K 0 Γ k , Γ T : = Γ × [ 0 , T ] , Γ T : = Γ × [ 0 , T ] , Q k , T : = Ω k × ( 0 , T ] for  k = 1 , , K 0 , Q k , T : = Ω k × ( 0 , T ] for  k = 1 , , K K 0 + 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_Equc_HTML.gif

We see that Γ T = Γ T Γ T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq35_HTML.gif.

C α ( Q ¯ T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq36_HTML.gif is the spaces of Hölder continuous in Q ¯ T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq37_HTML.gif with exponent α ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq38_HTML.gif. W 2 1 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq39_HTML.gif and W 2 1 , 1 ( Q T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq40_HTML.gif are the Hilbert spaces with scalar products ( v , w ) W 2 1 ( Ω ) = Ω ( v w + v x i w x i ) d x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq41_HTML.gif and ( v , w ) W 2 1 , 1 ( Q T ) = Q T ( v w + v t w t + v x i w x i ) d x d t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq42_HTML.gif, respectively. Let
W 2 1 ( Ω ) : = { v W 2 1 ( Ω ) , v | x Ω = 0 } , W 2 1 , 1 ( Q T ) : = { v W 2 1 , 1 ( Q T ) , v | ( x , t ) S T = 0 } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_Equd_HTML.gif

For the vector functions with N-components we denote the above function spaces by C α ( Q ¯ T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq43_HTML.gif, W 2 1 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq44_HTML.gif, W 2 1 , 1 ( Q T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq45_HTML.gif, W 2 1 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq46_HTML.gif and W 2 1 , 1 ( Q T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq47_HTML.gif, respectively.

Moreover, we recall the following.

Definition 2.1 (see [13, 15])

Write u in the split form
u = ( u l , [ u ] a l , [ u ] b l ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_Eque_HTML.gif
The vector function g ( , u ) : = ( g 1 ( , u ) , , g N ( , u ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq48_HTML.gif is said to be mixed quasimonotone in B R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq49_HTML.gif with index vector ( a 1 , , a N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq50_HTML.gif if for each l = 1 , , N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq51_HTML.gif, there exist nonnegative integers a l https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq52_HTML.gif, b l https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq53_HTML.gif, satisfying
a l + b l = N 1 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_Equf_HTML.gif

such that g l ( , u l , [ u ] a l , [ u ] b l ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq54_HTML.gif is nondecreasing in [ u ] a l https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq55_HTML.gif, and is nonincreasing in [ u ] b l https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq56_HTML.gif for all u B https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq57_HTML.gif.

The following hypotheses will be used in this paper:
  1. (H)
    (i) Ω and Γ k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq7_HTML.gif, k = 1 , , K 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq2_HTML.gif, are of C 2 + α 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq58_HTML.gif for some exponent α 0 ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq59_HTML.gif and there exist θ 0 ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq60_HTML.gif and ρ 0 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq61_HTML.gif such that for every open ball K ρ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq62_HTML.gif centered at x 0 Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq63_HTML.gif and radius ρ ρ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq64_HTML.gif,
    mes ( K ρ Ω ) ( 1 θ 0 ) mes K ρ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_Equg_HTML.gif
     
Assume that for each k = 1 , , K 0 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq26_HTML.gif,
Γ k : φ k ( x ) = 0 ( x Ω ¯ ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_Equh_HTML.gif
and
τ = 1 k Ω ¯ τ Γ k = { x : φ k ( x ) < 0 } Ω ¯ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_Equ3_HTML.gif
(2.1)
  1. (ii)
    Assume that
    { a i j l ( x , t , u l ) = a i j , k l ( x , t , u l ) , b j l ( x , t , u l ) = b j , k l ( x , t , u l ) , g l ( x , t , u ) = g k l ( x , t , u ) ( ( x , t ) Q k , T , u R N ) , k = 1 , , K 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_Equ4_HTML.gif
    (2.2)
     
where a i j , k l ( x , t , u l ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq65_HTML.gif and b j , k l ( x , t , u l ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq66_HTML.gif are defined on Q ¯ T × R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq67_HTML.gif, g k l ( x , t , u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq68_HTML.gif are defined on Q ¯ T × R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq69_HTML.gif, and all of them are allowed to be discontinuous on Γ T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq70_HTML.gif.
  1. (iii)
    There exist constant vectors M = ( M 1 , , M N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq71_HTML.gif and m = ( m 1 , , m N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq72_HTML.gif, m M https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq73_HTML.gif, such that
    { g k l ( x , t , M l , [ M ] a l , [ m ] b l ) 0 ( ( x , t ) Q T ) , g k l ( x , t , m l , [ m ] a l , [ M ] b l ) 0 ( ( x , t ) Q T ) , m l ψ l ( x , t ) M l ( ( x , t ) S T { Ω × { 0 } } ) , k = 1 , , K 0 , l = 1 , , N , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_Equ5_HTML.gif
    (2.3)
     
where a l https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq52_HTML.gif, b l https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq53_HTML.gif are all independent of k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq74_HTML.gif. Let
S : = { u C ( Q ¯ T ) : m u M } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_Equi_HTML.gif
The vector functions g k ( , u ) = ( g k 1 ( , u ) , , g k N ( , u ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq75_HTML.gif, k = 1 , , K 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq30_HTML.gif, are mixed quasimonotone in S https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq76_HTML.gif with the same index vector ( a 1 , , a N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq50_HTML.gif.
  1. (iv)
    For each k = 1 , , K 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq30_HTML.gif, k = 1 , , K K 0 + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq32_HTML.gif, l = 1 , , N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq51_HTML.gif, a i j , k l ( x , t , u l ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq65_HTML.gif, b j , k l ( x , t , u l ) C 1 + α 0 ( Q ¯ k , T × R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq77_HTML.gif ( i , j = 1 , , n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq78_HTML.gif), g k l ( x , t , u ) C 1 + α 0 ( Q ¯ k , T × S ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq79_HTML.gif. There exist a positive nonincreasing function ν ( θ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq80_HTML.gif and a positive nondecreasing function μ ( θ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq81_HTML.gif for θ [ 0 , + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq82_HTML.gif such that
    https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_Equ6_HTML.gif
    (2.4)
     
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_Equ7_HTML.gif
(2.5)
For each l = 1 , , N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq51_HTML.gif, ψ l ( x , t ) C α 0 ( Ξ ¯ × [ 0 , T ] ) W 2 1 , 1 ( Ξ × ( 0 , T ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq83_HTML.gif for some domain Ξ with Ω Ξ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq84_HTML.gif, ψ l ( x , 0 ) C 2 + α 0 ( Ω ¯ k ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq85_HTML.gif ( k = 1 , , K https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq6_HTML.gif), and the following compatibility condition on Γ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq86_HTML.gif holds:
[ a i j l ( x , 0 , ψ l ( x , 0 ) ) ψ x j l ( x , 0 ) ν i ( x ) ] Γ = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_Equ8_HTML.gif
(2.6)

Definition 2.2 A function u is said to be a solution of (1.1) if u possesses the following properties: (i) For some α ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq87_HTML.gif, u C α ( Q ¯ T ) C 2 , 1 ( Q k , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq88_HTML.gif, k = 1 , , K https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq6_HTML.gif. For any given Ω Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq22_HTML.gif and t ( 0 , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq89_HTML.gif, there exists α https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq90_HTML.gif, 0 < α < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq91_HTML.gif, such that u t C α ( Ω ¯ × [ t , T ] ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq92_HTML.gif and u x j L 2 ( Q T ) C α ( ( Ω ¯ Ω ¯ k ) × [ t , T ] ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq93_HTML.gif, k = 1 , , K https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq6_HTML.gif, j = 1 , , n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq94_HTML.gif; (ii) u satisfies the equations in (1.1) for ( x , t ) Q k , T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq95_HTML.gif, k = 1 , , K https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq6_HTML.gif, the diffraction conditions for ( x , t ) Γ T Q T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq96_HTML.gif and the parabolic boundary conditions for ( x , t ) S T ( Ω × { 0 } ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq97_HTML.gif.

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 S https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq76_HTML.gif.

2.2 An example

We next give an example satisfying the conditions in Hypothesis (H).

Example 2.1 In problem (1.1), let
n = 2 , φ = ( x 1 ) 2 + ( x 2 ) 2 100 , φ 1 = x 1 + ( x 2 ) 2 + 1 , φ 2 = x 1 ( x 2 ) 2 1 , φ 3 = ( x 1 4 ) 2 + ( x 2 ) 2 1 , Ω : φ = 0 , Γ 1 : φ 1 = 0 ( x 1 I 1 ) , Γ 2 : φ 2 = 0 ( x 1 I 2 ) , Γ 3 : φ 3 = 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_Equj_HTML.gif
where I 1 = [ ( 9 5 1 ) / 2 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq98_HTML.gif and I 2 = [ 1 , ( 9 5 1 ) / 2 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq99_HTML.gif, and let
Ω : φ < 0 , Ω 1 : φ < 0 , φ 1 < 0 , Ω 2 : φ < 0 , φ 1 > 0 , φ 2 < 0 , Ω 3 : φ < 0 , φ 2 > 0 , φ 3 > 0 , Ω 4 : φ 3 < 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_Equk_HTML.gif
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 Γ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq100_HTML.gif and Γ 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq101_HTML.gif intersect with Ω, and Γ 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq102_HTML.gif does not.
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_Fig1_HTML.jpg
Figure 1

The example of the domain and the interfaces for n = 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq103_HTML.gif .

For the coefficients of the equations and the boundary values in (1.1) we set
a i j l ( x , t , u l ) = { A k l E l ( u l ) , i = j , 0 , i j ( ( x , t ) Q k , T , u l R ) , k = 1 , 2 , 3 , 4 , i , j = 1 , 2 , b j l ( x , t , u l ) 0 ( ( x , t ) Q T , u l R ) , j = 1 , 2 , g l ( x , t , u ) = r k l u l f k l ( u ) ( ( x , t ) Q k , T , u R N ) , k = 1 , 2 , 3 , 4 , ψ l ( x , t ) o l , l = 1 , , N , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_Equl_HTML.gif
where
f k l ( u ) = 1 l = 1 N δ l , k l u l for  l = 1 , , N 1 , f k N ( u ) = 1 + l = 1 N 1 δ l , k N u l δ N , k N u N , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_Equm_HTML.gif

E l ( u l ) C 2 ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq104_HTML.gif with E l ( u l ) ν 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq105_HTML.gif, and ν 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq106_HTML.gif, A k l https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq107_HTML.gif, r k l https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq108_HTML.gif, δ l , k l https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq109_HTML.gif and o l https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq110_HTML.gif are all positive constants for k = 1 , 2 , 3 , 4 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq111_HTML.gif, l , l = 1 , , N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq112_HTML.gif.

Then
Γ 1 = Γ 1 , Γ 2 = Γ 2 , Ω 1 = Ω 1 , Ω 2 = Ω 2 , Ω 3 : φ < 0 , φ 2 > 0 , Γ 1 = Γ 3 , Ω 1 : φ < 0 , φ 3 > 0 , Ω 2 = Ω 4 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_Equn_HTML.gif
For each l = 1 , , N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq51_HTML.gif, let
a i j , k l ( x , t , u l ) = 0 ( ( x , t ) Q T , u l R ) , i j , i , j = 1 , 2 , k = 1 , 2 , 3 , a i i , k l ( x , t , u l ) = A k l E l ( u l ) ( ( x , t ) Q T , u l R ) , i = 1 , 2 , k = 1 , 2 , a i i , 3 l ( x , t , u l ) = { A 3 l E l ( u l ) ( ( x , t ) Q 1 , T , u l R ) , A 4 l E l ( u l ) ( ( x , t ) Q 2 , T , u l R ) , i = 1 , 2 , g k l ( x , t , u ) = r k l u l f k l ( u ) ( ( x , t ) Q T , u R N ) , k = 1 , 2 , g 3 l ( x , t , u ) = { r 3 l u l f 3 l ( u ) ( ( x , t ) Q 1 , T , u R N ) , r 4 l u l f 4 l ( u ) ( ( x , t ) Q 2 , T , u R N ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_Equo_HTML.gif
We find that these functions satisfy (2.2) and the hypothesis (iv) of (H). Set m = ( 0 , , 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq113_HTML.gif. Then the requirements on M in (2.3) become
1 δ l , k l M l 0 , M l o l , l = 1 , , N 1 . 1 + l = 1 N 1 δ l , k N M l δ N , k N M N 0 , M N o N . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_Equp_HTML.gif

It follows from these inequalities that there exist positive constant vector M, such that m and M satisfy (2.3). Furthermore, the vector functions g k ( , u ) = ( g k 1 ( , u ) , , g k N ( , u ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq75_HTML.gif, k = 1 , 2 , 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq114_HTML.gif, are mixed quasimonotone in S https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq76_HTML.gif with the same index vector ( 0 , , 0 , N 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq115_HTML.gif. The above arguments show that the conditions in Hypothesis (H) can be satisfied.

3 The proof of the existence theorem

3.1 Preliminaries

Lemma 3.1 The following statements hold true:
  1. (i)
    For any given x Ω ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq116_HTML.gif, if φ k 0 ( x ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq117_HTML.gif for some k 0 { 1 , , K 0 2 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq118_HTML.gif, then
    φ θ ( x ) < 0 for all θ { k 0 + 1 , , K 0 1 } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_Equq_HTML.gif
     
  2. (ii)
    There exists a positive number ε 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq119_HTML.gif such that for any given k { 2 , , K 0 1 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq120_HTML.gif, if 1 θ k 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq121_HTML.gif, then
    φ θ ( x ) ε 0 for all x { y : φ k ( y ) 0 } Ω ¯ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_Equr_HTML.gif
     

Proof By (2.1), if x Ω ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq116_HTML.gif and φ k 0 ( x ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq117_HTML.gif, then x τ = 1 k 0 Ω ¯ τ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq122_HTML.gif. Thus for each θ = k 0 + 1 , , K 0 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq123_HTML.gif, x τ = 1 θ Ω ¯ τ Γ θ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq124_HTML.gif. Again by (2.1) we get φ θ ( x ) < 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq125_HTML.gif. This proves the result in (i).

For any given k { 2 , , K 0 1 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq126_HTML.gif, if x Ω ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq116_HTML.gif and φ k ( x ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq127_HTML.gif, then it follows from (i) that φ θ ( x ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq128_HTML.gif for all θ { 1 , , k 1 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq129_HTML.gif. Since φ θ C 2 + α 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq130_HTML.gif, there exist positive constants ε k , θ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq131_HTML.gif such that
φ θ ( x ) ε k , θ for all  x { y : φ k ( y ) 0 } Ω ¯ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_Equs_HTML.gif

Hence, the conclusion in (ii) follows from the above relation by taking ε 0 : = min k , θ ε k , θ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq132_HTML.gif. □

For an arbitrary ε, 0 < ε < ε 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq133_HTML.gif, let s ε = s ε ( θ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq134_HTML.gif be smooth function with values between 0 and 1 such that | d d θ s ε ( θ ) | C / ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq135_HTML.gif for all θ R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq136_HTML.gif, s ε ( θ ) = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq137_HTML.gif for θ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq138_HTML.gif and s ε ( θ ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq139_HTML.gif for θ ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq140_HTML.gif. Define
z ε , k ( x ) : = { τ = 1 K 0 1 s ε ( φ τ ( x ) ) ( x Ω ¯ ) , k = 1 , ϑ = 1 K 0 1 [ 1 s ε ( φ ϑ ( x ) ) ] ( x Ω ¯ ) , k = K 0 , τ = k K 0 1 s ε ( φ τ ( x ) ) ϑ = 1 k 1 [ 1 s ε ( φ ϑ ( x ) ) ] ( x Ω ¯ ) , k = 2 , , K 0 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_Equ9_HTML.gif
(3.1)
Lemma 3.2 z ε , k ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq141_HTML.gif, k = 1 , , K 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq30_HTML.gif, are smooth functions with values between 0 and 1, and possess the property
k = 1 K 0 z ε , k ( x ) = 1 ( x Ω ¯ ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_Equ10_HTML.gif
(3.2)
Let functions η k ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq142_HTML.gif, k = 1 , , K 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq30_HTML.gif, be defined on Ω ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq143_HTML.gif, and let
η ε ( x ) = k = 1 K 0 η k ( x ) z ε , k ( x ) ( x Ω ¯ ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_Equ11_HTML.gif
(3.3)
Then for any x Ω ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq116_HTML.gif,
η ε ( x ) = { η 1 ( x ) 0 if φ 1 ( x ) 0 , η K 0 ( x ) if φ K 0 1 ( x ) ε , η k ( x ) , if φ k 1 ( x ) ε and φ k ( x ) 0 for some k { 2 , , K 0 1 } , η k 1 ( x ) s ε ( φ k 1 ( x ) ) + η k ( x ) [ 1 s ε ( φ k 1 ( x ) ) ] if 0 < φ k 1 ( x ) < ε for some k { 2 , , K 0 1 } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_Equ12_HTML.gif
(3.4)

Proof Since (3.2) is a special case of (3.4) with η k ( x ) 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq144_HTML.gif for all k { 1 , , K 0 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq145_HTML.gif, we only prove (3.4).

Case 1. If φ 1 ( x ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq146_HTML.gif, then the conclusion of (i) in Lemma 3.1 implies that φ k ( x ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq147_HTML.gif and s ε ( φ k ( x ) ) = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq148_HTML.gif for all k { 1 , , K 0 1 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq149_HTML.gif. (3.1) yields that z ε , 1 ( x ) = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq150_HTML.gif and z ε , k ( x ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq151_HTML.gif for k 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq152_HTML.gif. These, together with (3.3), imply that η ε ( x ) = η 1 ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq153_HTML.gif.

Case 2. If φ K 0 1 ( x ) ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq154_HTML.gif, then the conclusion of (ii) in Lemma 3.1 shows that φ k ( x ) ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq155_HTML.gif and s ε ( φ k ( x ) ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq156_HTML.gif for all k { 1 , , K 0 1 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq149_HTML.gif. Hence, z ε , K 0 ( x ) = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq157_HTML.gif and z ε , k ( x ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq151_HTML.gif for all k { 1 , , K 0 1 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq158_HTML.gif. Again by (3.3) we get η ε ( x ) = η K 0 ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq159_HTML.gif.

Case 3. If φ k ( x ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq147_HTML.gif and φ k 1 ( x ) ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq160_HTML.gif for some k { 2 , , K 0 1 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq126_HTML.gif, then Lemma 3.1 yields that φ τ ( x ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq161_HTML.gif, s ε ( φ τ ( x ) ) = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq162_HTML.gif for all τ { k , , K 0 1 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq163_HTML.gif, and that φ τ ( x ) ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq164_HTML.gif, s ε ( φ τ ( x ) ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq165_HTML.gif for all τ { 1 , , k 1 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq166_HTML.gif. Hence, z ε , k ( x ) = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq167_HTML.gif and z ε , τ ( x ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq168_HTML.gif for τ k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq169_HTML.gif. Therefore, η ε ( x ) = η k ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq170_HTML.gif.

Case 4. If 0 < φ k 1 ( x ) < ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq171_HTML.gif for some k { 2 , , K 0 1 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq126_HTML.gif, then it follows from Lemma 3.1 that φ τ ( x ) > ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq172_HTML.gif and s ε ( φ τ ( x ) ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq173_HTML.gif for all τ { 1 , , k 2 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq174_HTML.gif, and that φ k ( x ) < 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq175_HTML.gif. Again by the conclusion of (i) in Lemma 3.1 we have φ τ ( x ) < 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq176_HTML.gif and s ε ( φ τ ( x ) ) = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq177_HTML.gif for all τ { k , , K 0 1 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq178_HTML.gif. Hence, z ε , k ( x ) = 1 s ε ( φ k 1 ( x ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq179_HTML.gif, z ε , k 1 ( x ) = s ε ( φ k 1 ( x ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq180_HTML.gif and z ε , τ ( x ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq181_HTML.gif for τ k , k 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq182_HTML.gif. Thus, η ε ( x ) = η k 1 ( x ) s ε ( φ k 1 ( x ) ) + η k ( x ) [ 1 s ε ( φ k 1 ( x ) ) ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq183_HTML.gif. □

3.2 The approximation problem of (1.1)

In this subsection, we construct a problem to approximate (1.1).

For each l = 1 , , N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq51_HTML.gif, let
{ a i j ε l = a i j ε l ( x , t , u l ) : = k = 1 K 0 a i j , k l ( x , t , u l ) z ε , k ( x ) , b j ε l = b j ε l ( x , t , u l ) : = k = 1 K 0 b j , k l ( x , t , u l ) z ε , k ( x ) , g ε l = g ε l ( x , t , u ) : = k = 1 K 0 g k l ( x , t , u ) z ε , k ( x ) ( ( x , t ) Q T ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_Equ13_HTML.gif
(3.5)
It follows from hypothesis (iv) of (H), (3.2) and (3.5) that a i j ε l ( x , t , u l ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq184_HTML.gif, b j ε l ( x , t , u l ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq185_HTML.gif are in C 1 + α 0 ( Q ¯ k , T × R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq186_HTML.gif ( i , j = 1 , , n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq78_HTML.gif), g ε l ( x , t , u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq187_HTML.gif is in C 1 + α 0 ( Q ¯ k , T × S ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq188_HTML.gif ( k = 1 , , K K 0 + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq32_HTML.gif), the vector function g ε ( , u ) = ( g ε 1 ( , u ) , , g ε N ( , u ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq189_HTML.gif is mixed quasimonotone in S https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq76_HTML.gif with index vector ( a 1 , , a N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq50_HTML.gif, and
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_Equ14_HTML.gif
(3.6)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_Equ15_HTML.gif
(3.7)

We note that the functions a i j ε l ( x , t , u l ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq184_HTML.gif, b j ε l ( x , t , u l ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq185_HTML.gif and g ε l ( x , t , u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq187_HTML.gif are continuous on Γ T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq190_HTML.gif, and are allowed to be discontinuous on Γ T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq70_HTML.gif.

For each k = 1 , , K K 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq28_HTML.gif, there exists Ω τ k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq191_HTML.gif such that Γ k Ω τ k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq192_HTML.gif. Take two subdomains B k , 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq193_HTML.gif, B k , 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq194_HTML.gif satisfying Γ k B k , 1 B k , 2 Ω τ k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq195_HTML.gif. Let λ k = λ k ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq196_HTML.gif be an arbitrary smooth function taking values in [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq197_HTML.gif such that λ k = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq198_HTML.gif for x Ω τ k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq199_HTML.gif and λ k = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq200_HTML.gif for x B k , 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq201_HTML.gif. Set
ψ ε l = ψ ε l ( x , t ) : = | x y | ε ω ( | x y | ) ( 1 k = 1 K K 0 λ k ( y ) ) ψ l ( y , t ) d y + k = 1 K K 0 λ k ( x ) ψ l ( x , t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_Equ16_HTML.gif
(3.8)
with a sufficiently smooth nonnegative averaging kernel ω ( | ξ | ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq202_HTML.gif that is equal to zero for | ξ | 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq203_HTML.gif and is such that | ξ 1 ω ( ξ ) d ξ = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq204_HTML.gif. Then from the hypothesis (iv) of (H) and [[1], Chapter II] we know that for each l = 1 , , N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq51_HTML.gif, ψ ε l ( x , t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq205_HTML.gif is in C α 0 ( Q ¯ T ) W 2 1 , 1 ( Q T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq206_HTML.gif, ψ ε l ( x , 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq207_HTML.gif is in C 2 + α 0 ( Ω ¯ k ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq208_HTML.gif ( k = 1 , , K K 0 + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq32_HTML.gif), ψ ε l ψ l https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq209_HTML.gif in C α 0 ( Q ¯ T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq210_HTML.gif and ψ ε l ψ l https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq209_HTML.gif in W 2 1 , 1 ( Q T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq211_HTML.gif. Thus,
ψ ε l ( x , t ) C α 0 ( Q ¯ T ) + ψ ε l W 2 1 , 1 ( Q T ) μ 1 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_Equ17_HTML.gif
(3.9)
where μ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq212_HTML.gif is a positive constant, independent of ε. Furthermore, (3.4), (3.5) and (3.8) show that for small enough ε,
a i j ε l ( x , t , u l ) = a i j , τ k l ( x , t , u l ) , ψ ε l ( x , t ) = ψ l ( x , t ) ( ( x , t ) B k , 1 × [ 0 , T ] ) , k = 1 , , K K 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_Equt_HTML.gif
These, together with (2.6), imply that
[ a i j ε ( x , 0 , ψ ε l ( x , 0 ) ) ψ ε x j l ( x , 0 ) ν i ] Γ = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_Equ18_HTML.gif
(3.10)
For any given ε, 0 < ε < ε 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq133_HTML.gif, consider the approximation diffraction problem of (1.1)
{ u t l L ε l ( u l ) = g ε l ( x , t , u ) ( ( x , t ) Q T ) , [ u l ] Γ T = 0 , [ a i j ε l ( x , t , u l ) u x j l ν i ( x ) ] Γ T = 0 , u l = ψ ε l ( x , t ) ( ( x , t ) S T { Ω × { 0 } } ) , l = 1 , , N , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_Equ19_HTML.gif
(3.11)
where
L ε l ( u l ) : = d d x i ( a i j ε l ( x , t , u l ) u x j l ) + b j ε l ( x , t , u l ) u x j l . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_Equu_HTML.gif

We note that the interfaces in (3.11) are Γ k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq213_HTML.gif ( k = 1 , , K K 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq28_HTML.gif) which do not intersect with Ω. In view of (3.10), the compatibility condition on Γ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq86_HTML.gif holds.

Proposition 3.1 Problem (3.11) has a unique piecewise classical solution u ε = u ε ( x , t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq214_HTML.gif in S https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq76_HTML.gif possessing the following properties:
u ε C α ( Q ¯ T ) , u ε t C α , α / 2 ( Q ¯ T ) , u ε x j C α , α / 2 ( Q ¯ k , T ) ( α ( 0 , 1 ) ) , u ε x j t L 2 ( Q T ) , u ε x i x j C ( Q k , T ) , k = 1 , , K K 0 + 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_Equ20_HTML.gif
(3.12)

Proof Problem (3.11) is a special case of [[13], problem (1.1)] without time delays. Formulas (2.3) and (3.5) show that u ˜ = M https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq215_HTML.gif, u ˆ = m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq216_HTML.gif are a pair of bounded and coupled weak upper and lower solutions of (3.11) in the sense of [[13], Definition 2.2]. We find that the conditions of [[13], Theorem 4.1] are all fulfilled. Then from [[13], Theorem 4.1], we obtain that problem (3.11) has a unique piecewise classical solution u ε = u ε ( x , t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq214_HTML.gif in S https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq76_HTML.gif possessing the properties in (3.12). □

3.3 The uniform estimates of u ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq217_HTML.gif

In the following discussion, let K ρ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq62_HTML.gif be an arbitrary open ball of radius ρ with center at x 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq218_HTML.gif, and let Q ρ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq219_HTML.gif be an arbitrary cylinder of the form K ρ × [ t 0 ρ 2 , t 0 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq220_HTML.gif.

For each l = 1 , , N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq51_HTML.gif, consider the equality t 0 t Ω [ u ε t l L ε l ( u ε l ) ] η l d x d t = t 0 t Ω g ε l ( x , t , u ε ) η l d x d t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq221_HTML.gif for any function η l = η l ( x , t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq222_HTML.gif from W 2 1 , 1 ( Q T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq223_HTML.gif with ess sup Q T | η l | < https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq224_HTML.gif and for any t 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq225_HTML.gif, t from [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq226_HTML.gif. In view of u ε S https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq227_HTML.gif, it follows from (3.6), (3.7), (3.9) and the formula of integration by parts that
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_Equ21_HTML.gif
(3.13)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_Equ22_HTML.gif
(3.14)
Similarly, for any ϕ l W 2 1 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq228_HTML.gif and for every t [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq229_HTML.gif we get
Ω a i j ε l ( x , t , u ε l ) u ε x j l ϕ x i l d x = Ω [ u ε t l b j ε l ( x , t , u ε l ) u ε x j l + g ε l ( x , t , u ε ) ] ϕ l d x d t . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_Equ23_HTML.gif
(3.15)
Lemma 3.3 There exist constants α 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq230_HTML.gif ( 0 < α 1 < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq231_HTML.gif) and C depending only on M 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq232_HTML.gif ( : = max ( | M | , | m | ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq233_HTML.gif), ρ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq234_HTML.gif, θ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq235_HTML.gif, α 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq236_HTML.gif, ν ( M 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq237_HTML.gif, μ ( M 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq238_HTML.gif and μ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq212_HTML.gif, independent of ε, such that
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_Equ24_HTML.gif
(3.16)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_Equ25_HTML.gif
(3.17)

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 η l = u ε l ψ ε l https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq239_HTML.gif in (3.14) and using Cauchy’s inequality, we can obtain (3.17). □

Lemma 3.4 For any given k 1 { 1 , , K 0 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq240_HTML.gif, let D 1 Ω k 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq241_HTML.gif and t ( 0 , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq89_HTML.gif. Then there exist positive constants α 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq242_HTML.gif ( 0 < α 2 < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq243_HTML.gif) and C ( d 1 , t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq244_HTML.gif depending only on d 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq245_HTML.gif ( : = dist ( D 1 , Ω k 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq246_HTML.gif), t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq247_HTML.gif and the parameters M 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq232_HTML.gif, ρ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq234_HTML.gif, θ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq235_HTML.gif, α 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq236_HTML.gif, ν ( M 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq237_HTML.gif, μ ( M 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq238_HTML.gif and μ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq212_HTML.gif, independent of ε, such that for any Ω k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq31_HTML.gif satisfying D 1 Ω k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq248_HTML.gif,
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_Equ26_HTML.gif
(3.18)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_Equ27_HTML.gif
(3.19)
For any given k { 1 , , K } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq249_HTML.gif, let Ω Ω k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq250_HTML.gif and t ( 0 , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq251_HTML.gif. Then there exist positive constants α 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq252_HTML.gif ( 0 < α 3 < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq253_HTML.gif) and C ( d , t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq254_HTML.gif depending only on d https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq255_HTML.gif ( : = dist ( Ω , Ω k ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq256_HTML.gif), t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq257_HTML.gif and the parameters M 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq232_HTML.gif, ρ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq234_HTML.gif, θ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq235_HTML.gif, α 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq236_HTML.gif, ν ( ( M 0 ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq258_HTML.gif, μ ( M 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq238_HTML.gif and μ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq212_HTML.gif, such that
u ε l C 2 + α 3 , 1 + α 3 / 2 ( Ω ¯ × [ t , T ] ) C ( d , t ) , l = 1 , , N . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_Equ28_HTML.gif
(3.20)
Proof Choose a subdomain B satisfying D 1 B Ω k 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq259_HTML.gif. (3.4) and (3.5) show that for small enough ε,
{ a i j ε l ( x , t , u l ) = a i j , k 1 l ( x , t , u l ) , b j ε l ( x , t , u l ) = b j , k 1 l ( x , t , u l ) , g ε l ( x , t , u ) = g k 1 l ( x , t , u ) ( ( x , t ) B × ( 0 , T ] ) , l = 1 , , N . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_Equ29_HTML.gif
(3.21)

Then the same proofs as those of [[13], formulas (3.30) and (3.31)] give (3.18) and (3.19). If Ω Ω k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq250_HTML.gif, then Ω Ω k 1 Ω k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq260_HTML.gif for some k 1 { 1 , , K 0 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq240_HTML.gif, k { 1 , , K K 0 + 1 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq261_HTML.gif. Hence, the conclusion in (3.20) follows from (3.18), (3.19), (3.21) and the same argument as that for [[13], formula (3.37)]. □

In the rest of this subsection, let k 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq262_HTML.gif be an arbitrary fixed number in { 1 , , K 0 1 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq263_HTML.gif, and let D 2 Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq264_HTML.gif be an arbitrary fixed subdomain satisfying D 2 Γ k 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq265_HTML.gif, D ¯ 2 ( Γ k 2 1 Γ k 2 + 1 ) = https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq266_HTML.gif and D ¯ 2 Γ = https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq267_HTML.gif. We next investigate the uniform estimates in the neighborhood of Γ k 2 D ¯ 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq268_HTML.gif. Let x 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq218_HTML.gif be any point of Γ k 2 D ¯ 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq269_HTML.gif. [[2], Chapter 3, Section 16] and [13] show that there exists a ball K ρ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq62_HTML.gif with center at x 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq218_HTML.gif such that we can straighten Γ k 2 K ρ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq270_HTML.gif out by introducing a local coordinate system y = y ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq271_HTML.gif. Our assumptions concerning Γ imply that we can divide Γ k 2 D ¯ 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq272_HTML.gif 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 Γ k 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq273_HTML.gif lies in the plane x n = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq274_HTML.gif. Then by (3.4), when ( x , t ) D 2 × [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq275_HTML.gif the coefficients of problem (3.11) can be represented in the form
{ a i j ε l ( x , t , u l ) = a i j , k 2 l ( x , t , u l ) s ε ( x n ) + a i j , k 2 + 1 l ( x , t , u l ) [ 1 s ε ( x n ) ] , b j ε l ( x , t , u l ) = b j , k 2 l ( x , t , u l ) s ε ( x n ) + b j , k 2 + 1 l ( x , t , u l ) [ 1 s ε ( x n ) ] , g ε l ( x , t , u ) = g k 2 l ( x , t , u ) s ε ( x n ) + g k 2 + 1 l ( x , t , u ) [ 1 s ε ( x n ) ] , l = 1 , , N , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_Equ30_HTML.gif
(3.22)
and the diffraction conditions on Γ T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq276_HTML.gif in problem (1.1) can be represented in the form
[ u l ] Γ T = 0 , [ a n j l ( x , t , u l ) u x j l ] Γ T = 0 , l = 1 , , N . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_Equ31_HTML.gif
(3.23)
Lemma 3.5 Let t ( 0 , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq277_HTML.gif. Then there exist positive constants α 4 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq278_HTML.gif ( 0 < α 4 < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq279_HTML.gif) and C ( d 2 , t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq280_HTML.gif depending only on d 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq281_HTML.gif ( : = min { dist ( D 2 , Ω ) , dist ( D 2 , Γ k 2 1 Γ k 2 + 1 ) , dist ( D 2 , Γ ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq282_HTML.gif), t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq247_HTML.gif, and the parameters M 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq232_HTML.gif, ρ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq234_HTML.gif, θ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq235_HTML.gif, α 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq236_HTML.gif, ν ( M 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq237_HTML.gif, μ ( M 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq238_HTML.gif and μ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq212_HTML.gif, independent of ε, such that
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_Equ32_HTML.gif
(3.24)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_Equ33_HTML.gif
(3.25)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_Equ34_HTML.gif
(3.26)
Proof It follows from (3.22) and Hypothesis (H) that
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_Equ35_HTML.gif
(3.27)
and from the equations in (3.11) that
| d d x n ( a n j ε l ( x , t , u ε l ) u ε x j l ) | C ( | u ε t l | + s = 1 n 1 j = 1 n | u ε x j x s l | + | u ε x l | 2 + 1 ) ( ( x , t ) D ¯ 2 × [ 0 , T ] ) , l = 1 , , N . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_Equ36_HTML.gif
(3.28)

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 Arzela-Ascoli theorem it follows that we can find a subsequence (we retain the same notation for it) { u ε } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq283_HTML.gif such that { u ε } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq284_HTML.gif converges in C ( Q ¯ T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq285_HTML.gif to u and { u ε x j } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq286_HTML.gif converges weakly in L 2 ( Q T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq287_HTML.gif to u x j https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq288_HTML.gif for each j = 1 , , n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq94_HTML.gif. Then u C α 1 ( Q ¯ T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq289_HTML.gif and u x j L 2 ( Q T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq290_HTML.gif. Furthermore, the parabolic boundary conditions for u ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq217_HTML.gif in (3.11) imply that u satisfies the parabolic boundary conditions in (1.1).

For any given k { 1 , , K } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq249_HTML.gif, and for any Ω Ω k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq250_HTML.gif, t ( 0 , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq251_HTML.gif, (3.20) yields that there exists a subsequence { u ε } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq291_HTML.gif (denoted by { u ε } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq283_HTML.gif still) such that { u ε } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq283_HTML.gif converges in C 2 , 1 ( Ω ¯ × [ t , T ] ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq292_HTML.gif to u. By letting ε 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq293_HTML.gif, from (3.21) and the equations u ε t l L ε l ( u ε l ) = g ε l ( x , t , u ε ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq294_HTML.gif in (3.11) we get that
u t l L l ( u l ) = g l ( x , t , u ) ( ( x , t ) Ω × [ t , T ] ) , l = 1 , , N . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_Equv_HTML.gif

Since Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq295_HTML.gif and t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq257_HTML.gif are arbitrary, then u satisfies the equations in (3.11) for ( x , t ) Q k , T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq95_HTML.gif.

For any given k 1 { 1 , , K 0 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq240_HTML.gif and for any D 1 Ω k 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq241_HTML.gif, t ( 0 , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq89_HTML.gif, we see from (3.18), (3.19) that there exists a subsequence { u ε } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq291_HTML.gif (denoted by { u ε } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq283_HTML.gif still) such that for each j = 1 , , n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq94_HTML.gif and for any Ω k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq31_HTML.gif satisfying D 1 Ω k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq296_HTML.gif, { u ε x j } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq286_HTML.gif converges in C ( ( D 1 Ω k ¯ ) × [ t , T ] ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq297_HTML.gif to u x j https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq288_HTML.gif, and { u ε t } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq298_HTML.gif converges in C ( D ¯ 1 × [ t , T ] ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq299_HTML.gif to u t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq300_HTML.gif. Hence
u x j C α 2 ( ( D 1 Ω k ¯ ) × [ t , T ] ) , u t C α 2 ( D ¯ 1 × [ t , T ] ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_Equ37_HTML.gif
(3.29)
By letting ε 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq293_HTML.gif we conclude from (3.21) and the diffraction conditions on Γ T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq301_HTML.gif for u ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq217_HTML.gif in (3.11) that
[ u l ] Γ T Q T = 0 , [ a i j l ( x , t , u l ) u x j l ν i ( x ) ] Γ T Q T = 0 , l = 1 , , N . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_Equ38_HTML.gif
(3.30)
For any given k 2 { 1 , , K 0 1 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq302_HTML.gif and D 2 Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq264_HTML.gif satisfying D 2 Γ k 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq265_HTML.gif, D ¯ 2 ( Γ k 2 1 Γ k 2 + 1 ) = https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq266_HTML.gif and D ¯ 2 Γ = https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq267_HTML.gif, the estimates (3.24)-(3.26) imply that for any given t ( 0 , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq89_HTML.gif there exists a subsequence { u ε } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq291_HTML.gif (denoted by { u ε } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq283_HTML.gif still) such that for each s = 1 , , n 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq303_HTML.gif, l = 1 , , N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq51_HTML.gif,
u ε x s l u x s l , u ε t l u t l , ϖ ε , n l = a n j ε l ( x , t , u ε l ) u ε x j l ϖ l in  C ( D ¯ 2 × [ t , T ] ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_Equ39_HTML.gif
(3.31)
Then
u x s l , u t l , ϖ l C α 4 ( D ¯ 2 × [ t , T ] ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_Equ40_HTML.gif
(3.32)
We next show that ϖ l = a n j l ( x , t , u l ) u x j l https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq304_HTML.gif. For any η = η ( x , t ) L 2 ( D 2 × ( t , T ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq305_HTML.gif,
t T D 2 [ a n j ε l ( x , t , u ε l ) u ε x j l a n j l ( x , t , u l ) u x j l ] η d x d t = t T D 2 ( a n j ε l ( x , t , u ε l ) a n j ε l ( x , t , u l ) ) u ε x j l η d x d t + t T D 2 ( a n j ε l ( x , t , u l ) a n j l ( x , t , u l ) ) u ε x j l η d x d t + t T D 2 a n j l ( x , t , u l ) ( u ε x j l u x j l ) η d x d t : = I ε , 1 l + I ε , 2 l + I ε , 3 l . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_Equw_HTML.gif
By (3.27), (3.17), we get
| I ε , 1 l | C ( u ε l u l ) η L 2 ( D 2 × ( t , T ) ) u ε x j l L 2 ( D 2 × ( t , T ) ) C ( u ε l u l ) η L 2 ( D 2 × ( t , T ) ) 0 as  ε 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_Equx_HTML.gif
and by (2.2), (3.22),
| I ε , 2 l | = | t T D 2 { x | 0 x n ε } ( a n j , k 2 l ( x , t , u l ) a n j , k 2 + 1 l ( x , t , u l ) ) s ε ( x n ) u ε x j l η d x d t | C u ε x j l L 2 ( D 2 × ( 0 , T ) ) { t T D 2 { x | 0 x n ε } η 2 d x d t } 1 / 2 C { t T D 2 { x | 0 x n ε } η 2 d x d t } 1 / 2 0 as  ε 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_Equy_HTML.gif
Since { u ε x j } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq286_HTML.gif converges weakly in L 2 ( Q T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq287_HTML.gif to u x j https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq288_HTML.gif for each j = 1 , , n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq94_HTML.gif, then I ε , 3 l 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq306_HTML.gif as ε 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq293_HTML.gif. Hence, ϖ ε , n l = a n j ε l ( x , t , u ε l ) u ε x j l https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq307_HTML.gif converges weakly in L 2 ( D 2 × ( t , T ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq308_HTML.gif to a n j l ( x , t , u l ) u x j l https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq309_HTML.gif for each j = 1 , , n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq94_HTML.gif. This, together with (3.31), implies that
ϖ l = a n j l ( x , t , u l ) u x j l C α 4 ( D ¯ 2 × [ t , T ] ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_Equ41_HTML.gif
(3.33)

and u satisfies the diffraction conditions on Γ T Q T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq310_HTML.gif in (3.23).

In view of (3.30) u satisfies the diffraction conditions on Γ T Q T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq311_HTML.gif in (1.1). Furthermore, (3.29), (3.32) and (3.33) imply that for any k { 1 , , K } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq249_HTML.gif, Ω Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq22_HTML.gif,
u x j C α ( ( Ω Ω k ¯ ) × [ t , T ] ) , u t C α ( Ω ¯ × [ t , T ] ) , j = 1 , , n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_Equz_HTML.gif

for some α ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-99/MediaObjects/13661_2013_Article_353_IEq312_HTML.gif. Therefore, u is a solution of (1.1). This completes the proof of Theorem 2.1.

Declarations

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 (CSYXM12-06).

Authors’ Affiliations

(1)
Department of Mathematics, Chengdu Normal University

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