Báo cáo hóa học: " Research Article Systems of Quasilinear Parabolic Equations with Discontinuous Coefficients and Continuous Delays" docx
Bạn đang xem bản rút gọn của tài liệu. Xem và tải ngay bản đầy đủ của tài liệu tại đây (637.89 KB, 25 trang )
Hindawi Publishing Corporation
Advances in Difference Equations
Volume 2011, Article ID 925173, 25 pages
doi:10.1155/2011/925173
Research Article
Systems of Quasilinear Parabolic Equations with
Discontinuous Coefficients and Continuous Delays
Qi-Jian Tan
Department of Mathematics, Sichuan College of Education, Chengdu 610041, China
Correspondence should be addressed to Qi-Jian Tan,
Received 24 December 2010; Accepted 3 March 2011
Academic Editor: Jin Liang
Copyright q 2011 Qi-Jian Tan. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
This paper is concerned with a weakly coupled system of quasilinear parabolic equations where
the coefficients are allowed to be discontinuous and the reaction functions may depend on
continuous delays. By the method of upper and lower solutions and the associated monotone
iterations and by difference ratios method and various estimates, we obtained the existence and
uniqueness of the global piecewise classical solutions under certain conditions including mixed
quasimonotone property of reaction functions. Applications are given to three 2-species Volterra-
Lotka models with discontinuous coefficients and continuous delays.
1. Introduction
Reaction-diffusion equations with time delays have been studied by many researchers
see 1–8 and references therein. However, all of the discussions in the literature are
devoted to the equations with continuous coefficients. In this paper, we consider a weakly
coupled system of quasilinear parabolic equations where t he coefficients are allowed to
be discontinuous and the reaction functions may depend on continuous infinite or finite
delays.
To describe the problem, we first introduce some notations. Let Ω be a bounded
domain with the boundary ∂Ω in R
n
n ≥ 1. Suppose that Ω consists of a finite number
of domains Ω
k
k 1, ,K separated by Γ
k
, where Γ
k
, k
1, ,K
, are surfaces which
do not intersect with each other and with ∂Ω. Γ : ∪Γ
k
and
−→
n is the normal to Γ. The symbol
v
Γ×0,∞
denotes the jump in the function v as it crosses Γ ×0, ∞. For any vector function
u u
1
, ,u
N
, we write u
l
t
: ∂u
l
/∂t, u
l
x
i
: ∂u
l
/∂x
i
, u
l
x
:u
l
x
1
, ,u
l
x
n
, l 1, ,N,
i 1, ,n.
2 Advances in Difference Equations
In this paper, we consider the following reaction-diffusion system:
u
l
t
−L
l
u
l
g
l
x, t, u,J ∗ u
x ∈ Ω,t>0
,
u
l
Γ×0,∞
0,
⎡
⎣
n
i,j1
a
l
ij
x, t, u
l
u
l
x
j
cos
−→
n,x
i
⎤
⎦
Γ×0,∞
0,
u
l
h
l
x, t
x ∈ ∂Ω,t≥ 0
,
u
l
x, t
ψ
l
x, t
x ∈ Ω,t∈ I
l
,l 1, ,N,
1.1
where
J ∗ u :
J
1
∗ u
1
, ,J
N
∗ u
N
,J
l
∗ u
l
:
I
l
∪0,t
J
l
x, t − s
u
l
x, s
ds,
1.2
L
l
u
l
:
n
i1
d
dx
i
⎛
⎝
n
j1
a
l
ij
x, t, u
l
u
l
x
j
⎞
⎠
n
j1
b
l
j
x, t, u
l
u
l
x
j
,
I
l
:
⎧
⎨
⎩
−∞, 0
for l 1, ,N
0
,
−r
l
, 0
for l N
0
1, ,N,
1.3
the expressions d/dx
i
a
l
ij
x, t, u
l
u
l
x
j
mean that
d
dx
i
a
l
ij
x, t, u
l
u
l
x
j
⎡
⎣
∂a
l
ij
x, t, u
l
∂x
i
∂a
l
ij
x, t, u
l
∂u
l
u
l
x
i
⎤
⎦
u
l
x
j
a
l
ij
x, t, u
l
u
l
x
j
x
i
, 1.4
N
0
is a nonnegative integer, and r
l
, l N
0
1, ,N, are positive constants.
The equations with discontinuous coefficients have been investigated extensively in
the literature see 9–16 and references therein. However, the discussions in these literature
are devoted either to scalar equations without time delays or to coupled system of equations
without time delays and with the restrictive conditions that the principal parts are the same
and the convection functions b
l
x, t, u,u
l
x
satisfy see 16
u
l
b
l
x, t, u, 0
≥−C
1
|
u
|
2
− C
2
x ∈
Ω
k
,t∈
0,T
, u ∈ R
N
,k 1, ,K, l 1, ,N.
1.5
In this paper we will extend the method of upper and lower solutions and the monotone
iteration scheme to reaction-diffusion system with discontinuous coefficients and continuous
delays and use these methods and the results of 15, 16 to prove the existence and uniqueness
of the piecewise classical solutions for 1.1 under hypothesis H in Section 2.
This paper is organized as follows. In the next section we will prove a weak
comparison principle and construct two monotone sequences. Section 3 is devoted to
Advances in Difference Equations 3
investigate the uniform estimates of the sequences. In Section 4 we prove the existence and
uniqueness of the piecewise classical solutions for 1.1. Applications of these results are
given in Section 5 to three 2-species Volterra-Lotka models with discontinuous coefficients
and continuous delays.
2. Two Monotone Sequences
The aim of this section is to prove a weak comparison principle and construct two monotone
sequences. In Section 4 we will show that these sequences converge to the unique solution of
1.1.
2.1. The Definitions, Hypotheses, and Weak Comparison Principle
In all that follows, pairs of indices i or j imply a summation from 1 to n. The symbol Ω
⊂⊂ Ω
means that Ω
⊂ Ω and distΩ
,∂Ω > 0. For any T>0, we set
Ω :Ω∪ ∂Ω, Ω
k
:Ω
k
∪ ∂Ω
k
,S
T
: ∂Ω ×
0,T
, Γ
T
:Γ×
0,T
,
D
T
:Ω×
0,T
,D
k,T
:Ω
k
×
0,T
, D
T
: Ω ×
0,T
, D
k,T
: Ω
k
×
0,T
,
D
T
: D
T
×···×D
T
N
, D
k,T
: D
k,T
×···×D
k,T
N
, D
T
: D
T
×···×D
T
N
,
Q
l
0
:Ω× I
l
,Q
l
k,0
:Ω
k
× I
l
, Q
l
0
: Ω × I
l
, Q
l
k,0
Ω
k
× I
l
,
Q
l
T
:Ω×
I
l
∪
0,T
,Q
l
k,T
:Ω
k
×
I
l
∪
0,T
,
Q
l
T
: Ω ×
I
l
∪
0,T
,
Q
T
: Q
1
T
×···×Q
N
T
, Q
k,T
: Q
1
k,T
×···×Q
N
k,T
,k 1, ,K, l 1, ,N.
2.1
Let |u| :
N
l1
u
l
2
1/2
, |u
l
x
| :
n
i1
u
l
x
i
2
1/2
, |u
l
xx
| :
n
i,j1
u
l
x
i
x
j
2
1/2
.
W
1,0
2
D
T
and W
1,1
2
D
T
are the Hilbert spaces with scalar products v, w
W
1,0
2
D
T
D
T
vw v
x
i
w
x
i
dx dt and v, w
W
1,1
2
D
T
D
T
vw v
t
w
t
v
x
i
w
x
i
dx dt, respectively.
◦
W
1,1
2
D
T
and
◦
W
1,0
2
D
T
are the sets of all functions in W
1,1
2
D
T
and W
1,0
2
D
T
that vanish
on S
T
in the sense of trace, respectively. For vector functions with N-components, we use the
notations
C
α
D
T
: C
α
D
T
×···×C
α
D
T
N
, W
1,1
2
DT
: W
1,1
2
DT
×···×W
1,1
2
DT
N
,
C
α
Q
T
: C
α
Q
1
T
×···×C
α
Q
N
T
.
2.2
In Section 3 the same notations are also used to denote the spaces of the vector functions with
2N-components. Similar notations are used for other function spaces and other domains.
4 Advances in Difference Equations
Definition 2.1 see 3, 5.Writeu, v in the split form
u
u
l
,
u
a
l
,
u
b
l
, v
v
c
l
,
v
d
l
. 2.3
The vector function g·, u, v :g
1
·, u, v, ,g
N
·, u, v is said to be mixed quasimonotone
in A ⊂ R
N
× R
N
if, for each l 1, ,N, there exist nonnegative integers a
l
, b
l
, c
l
,andd
l
satisfying
a
l
b
l
N − 1,c
l
d
l
N,
2.4
such that g
l
·,u
l
, u
a
l
, u
b
l
, v
c
l
, v
d
l
is nondecreasing in u
a
l
and v
c
l
, and is nonincreas-
ing in u
b
l
and v
d
l
for all u, v ∈ A.
Let
H
l
τ; v, η
:
D
τ
v
l
t
η
l
a
l
ij
x, t, v
l
v
l
x
j
η
l
x
i
b
l
j
x, t, v
l
v
l
x
j
η
l
dx dt, 2.5
where D
τ
:Ω× 0,τ.
Definition 2.2. A pair of functions
u u
1
, ,u
N
,
u u
1
, ,u
N
are called coupled weak
upper and lower solutions of 1.1 if i
u and
u are in C
α
0
Q
T
∩C
1α
0
D
k,T
k 1, ,K
for some α
0
∈ 0, 1, ii
u ≥
u and iii for any nonnegative vector function η η
1
, ,η
N
∈
◦
W
1,1
2
D
T
and any τ ∈ 0,T
H
l
τ;
u, η
≥
D
τ
g
l
x, t, u
l
,
u
a
l
,
u
b
l
,
J ∗
u
c
l
,
J ∗
u
d
l
η
l
dx dt,
H
l
τ;
u, η
≤
D
τ
g
l
x, t, u
l
,
u
a
l
,
u
b
l
,
J ∗
u
c
l
,
J ∗
u
d
l
η
l
dx dt,
u
l
≤ g
l
x, t
≤ u
l
x, t
∈ S
T
,
u
l
x, t
≤ ψ
l
x, t
≤ u
l
x, t
x, t
∈ Q
l
0
,l 1, ,N.
2.6
Throughout this paper the following hypotheses will be used.
Hi ∂Ω and Γ
k
, k 1, ,K
,areofC
2α
0
for some exponent α
0
∈ 0, 1, and there
exist positive numbers a
0
and θ
0
such that
mes
K
ρ
∩ Ω
≤
1 − θ
0
mes K
ρ
2.7
holds for any open ball K
ρ
with center on ∂Ω of radius ρ ≤ a
0
.
Advances in Difference Equations 5
ii There exist a pair of bounded and coupled weak upper and lower solutions
u,
u.
We set
S :
u ∈C
Q
T
:
u ≤ u ≤
u
, S
∗
:
w ∈C
Q
T
: J ∗
u ≤ w ≤ J ∗
u
,
S
∗l
:
w
l
∈ C
Q
l
T
: J
l
∗ u ≤ w
l
≤ J
l
∗ u
,l 1, ,N.
2.8
iii For each k 1, ,K, l 1, ,N, a
l
ij
x, t, u
l
,b
l
j
x, t, u
l
∈ C
1α
0
D
k,T
× Ri, j
1, ,n,g
l
x, t, u, v ∈ C
1α
0
D
k,T
×S×S
∗
, h
l
x, t ∈ C
2α
0
S
T
, ψ
l
x, 0 ∈
C
α
0
Ω ∩ C
2α
0
Ω
k
. There exist a positive nonincreasing function νθ,apositive
nondecreasing function μθ for θ ∈ 0, ∞, and a positive constant μ
1
such that
ν
u
l
n
i
1
ξ
2
i
≤
n
i
,j
1
a
l
i
j
x, t, u
l
ξ
i
ξ
j
≤ μ
u
l
n
i
1
ξ
2
i
,
2.9
a
l
ij
a
l
ji
,
a
l
ij
x, t, u
l
; b
l
j
x, t, u
l
≤ μ
u
l
,i,j 1, ,N, 2.10
g
l
x, t, u, v
C
1
D
k,T
×S×S
∗
≤ μ
1
, 2.11
h
l
C
2α
0
S
T
ψ
l
x, 0
C
α
0
Ω
ψ
l
x, 0
C
2α
0
Ω
k
≤ μ
1
.
2.12
iv For each l 1, ,N, J
l
x, t ∈ C
α
0
Ω × I
l
∗
∩ C
1α
0
D
k,T
,
J
l
x, t
≥ 0
x, t
∈ Ω × I
l
∗
,
I
l
∗
J
l
x, t
dt 1
x ∈ Ω
, 2.13
where I
l
∗
:0, ∞ for l 1, ,N
0
and I
l
∗
:0,r
l
for l N
0
1, ,N, ψ
l
x, t ∈
C
α
0
Q
l
0
,and
I
l
J
l
x, t − sψ
l
x, sds, J
l
∗ u
l
,J
l
∗ u
l
∈ C
1α
0
D
k,T
. There exists a
constant μ
2
such that
ψ
l
x, t
CQ
l
0
≤ μ
2
,
I
l
J
l
x, t − s
ψ
l
x, s
ds; J
l
∗ u
l
; J
l
∗ u
l
C
1α
0
D
k,T
≤ μ
2
.
2.14
v The vector function g·, u, vg
1
·, u, v, ,g
N
·, u, v is mixed quasimonotone
in S×S
∗
.
vi The following compatibility conditions hold:
h
l
x, 0
ψ
l
x, 0
x ∈ ∂Ω
,
a
l
ij
x, 0,ψ
l
x, 0
∂ψ
l
x, 0
∂x
j
cos
−→
n,x
i
Γ
0,l 1, ,N.
2.15
6 Advances in Difference Equations
The weak upper and lower solutions
u,
u in hypothesis H-ii will be used as the
initial iterations to construct two monotone convergent sequences.
Definition 2.3. A function u is called a piecewise classical solution of 1.1 if i u ∈C
α
Q
T
,
u
t
∈C
α,α/2
D
T
, u
x
j
∈C
α,α/2
D
k,T
for some α ∈ 0, 1, u
x
j
t
∈L
2
D
T
, j 1, ,n; and for any
given k, k 1, ,K, and any given Ω
⊂⊂ Ω
k
and t
∈ 0,T, there exists α
∈ 0, 1 such
that u
l
x
i
x
j
∈ C
α
,α
/2
Ω
× t
,T, i, j 1, ,n, l 1, ,N,andifii u satisfies pointwise the
equations in 1.1 for x, t ∈ D
k,T
, k 1, ,K, and satisfies pointwise the inner boundary
conditions in 1.1 on Γ
T
, the parabolic conditions on S
T
, and the initial conditions u
l
x, t
ψ
l
x, t in Q
l
0
.
To construct the monotone sequences, we next prove the weak comparison principle.
Lemma 2.4. Let functions a
l
ij
x, t, u
l
, b
l
j
x, t, u
l
, l 1, ,N, satisfy the conditions in hypothesis
(H).
i Assume that q
l
x, t, Y, Z ∈ C
1α
0
D
k,T
×S×S
∗
,l 1, ,N, and the vector function
q·, Y, Zq
1
·, Y, Z, ,q
N
·, Y, Z is mixed quasimonotone in S×S
∗
.Ifv, u ∈
C
D
T
∩W
1,1
∞
D
T
∩Sand if
H
l
τ; v, η
−
D
τ
q
l
x, t, v
l
,
v
a
l
,
u
b
l
,
J ∗ v
c
l
,
J ∗ u
d
l
η
l
dx dt
≤H
l
τ; u, η
−
D
τ
q
l
x, t, u
l
,
u
a
l
,
v
b
l
,
J ∗ u
c
l
,
J ∗ v
d
l
η
l
dx dt,
v
l
x, t
≤ u
l
x, t
x, t
∈ S
T
,
v
l
x, t
u
l
x, t
ψ
l
x, t
x, t
∈ Q
l
0
,l 1, ,N,
2.16
for any nonnegative bounded vector function η η
1
, ,η
N
∈
◦
W
1,1
2
D
T
and any τ ∈
0,T,thenv ≤ u for x, t ∈
D
T
.
ii If v, u ∈C
D
T
∩W
1,1
∞
D
T
and if
H
l
τ; v, η
D
τ
e
l
x, t
v
l
η
l
dx dt ≤H
l
τ; u, η
D
τ
e
l
x, t
u
l
η
l
dx dt,
v
l
x, t
≤ u
l
x, t
x, t
∈ S
T
,v
l
x, 0
≤ u
l
x, 0
x ∈ Ω
,l 1, ,N,
2.17
for any nonnegative bounded vector function η ∈
◦
W
1,1
2
D
T
,wheree
l
x, t,l 1, ,N,
are functions in C
D
k,T
k 1, ,K), then v ≤ u for x, t ∈ D
T
.
Advances in Difference Equations 7
Proof. We first prove part i of the lemma. Let w v − u, w
w
1
, ,w
N
:
maxw
1
, 0, ,maxw
N
, 0. Then w
l
0forx, t ∈ S
T
∪ Q
l
0
, l 1, ,N. Choosing
η w
in 2.16,weobtain
N
l1
H
l
τ; v, w
−H
l
τ; u, w
≤
N
l1
D
τ
q
l
x, t, v
l
,
v
a
l
,
u
b
l
,
J ∗ v
c
l
,
J ∗ u
d
l
− q
l
x, t, u
l
,
u
a
l
,
v
b
l
,
J ∗ u
c
l
,
J ∗ v
d
l
w
l
dx dt
N
l1
D
τ
⎡
⎣
E
l
1l
w
l
w
l
∈w
a
l
E
l
1l
w
l
w
l
∈w
b
l
E
l
1l
−w
l
J
l
∗w
l
∈
J∗w
c
l
E
l
2l
J
l
∗ w
l
J
l
∗w
l
∈
J∗w
d
l
E
l
2l
−J
l
∗ w
l
⎤
⎦
w
l
dx dt,
2.18
where
E
l
1l
1
0
∂q
l
x, t, Y
θ
, Z
θ
∂y
l
θ
dθ, E
l
2l
1
0
∂q
l
x, t, Y
θ
, Z
θ
∂z
l
θ
dθ,
Y
θ
, Z
θ
y
l
θ
,
Y
θ
a
l
,
Y
θ
b
l
,
Z
θ
c
l
,
Z
θ
d
l
: θ
v
l
,
v
a
l
,
u
b
l
,
J ∗ v
c
l
,
J ∗ u
d
l
1 − θ
u
l
,
u
a
l
,
v
b
l
,
J ∗ u
c
l
,
J ∗ v
d
l
.
2.19
Let us estimate the terms in 2.18. It follows from the mixed quasimonotone property
of q·, Y, Z, 2.13 and 2.14 that, for each l 1, ,N,
E
l
1l
w
l
≤ E
l
1l
w
l
for w
l
∈
w
a
l
, −E
l
1l
w
l
≤−E
l
1l
w
l
for w
l
∈
w
b
l
,
E
l
2l
J
l
∗ w
l
≤ E
l
2l
J
l
∗ w
l
≤ E
l
2l
J
l
∗
w
l
for J
l
∗ w
l
∈
J ∗ w
c
l
,
−E
l
2l
J
l
∗ w
l
≤−E
l
2l
J
l
∗ w
l
≤−E
l
2l
J
l
∗
w
l
for J
l
∗ w
l
∈
J ∗ w
d
l
,
2.20
E
l
1l
E
l
2l
≤ C
O
,l
1, ,N, 2.21
where O |u|
CD
T
|v|
CD
T
|J ∗ u|
CD
T
|J ∗ v|
CD
T
.Hereandbelowinthis
section, C··· denotes the constant depending only on μ
1
, μ
2
, and the quantities appearing
8 Advances in Difference Equations
in parentheses. Constant C in different expressions may be different. By hypothesis H- iv
and H
¨
older’s inequality, we have that
D
τ
J
l
∗
w
l
2
dxdt
D
τ
t
0
J
l
x, t − s
w
l
x, s
ds
2
dx dt
≤
D
τ
t
0
J
l
x, t − s
2
ds
t
0
w
l
x, s
2
ds
dx dt
≤ C
D
τ
τ
0
w
l
x, s
2
ds
dx dt
≤ Cτ
D
τ
w
l
x, t
2
dx dt, l
1, ,N,
2.22
and by 2.5, 2.9, 2.10, and Cauchy’s inequality, we have that
N
l1
H
l
τ; v, w
−H
l
τ; u, w
1
2
Ω
w
x, τ
2
dx
N
l1
D
τ
a
l
ij
x, t, v
l
w
l
x
j
a
l
ij
x, t, v
l
− a
l
ij
x, t, u
l
u
l
x
j
w
l
x
i
b
l
j
x, t, v
l
v
l
x
j
− b
l
j
x, t, u
l
u
l
x
j
w
l
dx dt
≥
1
2
Ω
w
x, τ
2
dx
ν
O
− ε
N
l1
D
τ
w
l
x
2
dx dt − C
O
1
D
τ
|
w
|
2
dx dt,
2.23
where O
1
|u|
CD
T
|v|
CD
T
N
l1
|u
l
x
|
L
∞
D
T
|v
l
x
|
L
∞
D
T
.
Setting ε νO/2 and substituting relations 2.20–2.23 into 2.18,weseethat
Ω
w
x, τ
2
dx
N
l1
D
τ
w
l
x
2
dx dt ≤ C
O, O
1
D
τ
|
w
|
2
|
J ∗ w
|
2
dx dt
≤ C
O, O
1
D
τ
|
w
|
2
dx dt.
2.24
Hence, we deduce the relation w
≡ 0 from this inequality with the use of Gronwall
inequality. Then, v ≤ u in
D
T
, and the proof of part i of the lemma is completed. The similar
argument gives the proof of part ii of the lemma.
Advances in Difference Equations 9
2.2. Construction of Monotone Sequences
In this subsection, we construct the monotone sequences. By hypothesis H-iii, for each
l 1, ,N, there exists
l
l
x, t ∈ C
2
D
k
k 1, ,K satisfying
l
x, t
≥ max
−
∂g
l
x, t, u, v
∂u
l
:
u, v
∈S×S
∗
. 2.25
Define
G
l
x, t, u, v
G
l
x, t, u
l
,
u
a
l
,
u
b
l
,
v
c
l
,
v
d
l
:
l
u
l
g
l
x, t, u
l
,
u
a
l
,
u
b
l
,
v
c
l
,
v
d
l
.
2.26
Since g·, ug
1
·, u, ,g
N
·, u is mixed quasimonotone in S×S
∗
, then, for any
u, v, u
∗
, v
∗
∈S×S
∗
, u, v ≤ u
∗
, v
∗
,
G
l
·,u
l
,
u
a
l
,
u
∗
b
l
,
v
c
l
,
v
∗
d
l
≤ G
l
·,u
∗l
,
u
∗
a
l
,
u
b
l
,
v
∗
c
l
,
v
d
l
. 2.27
It is obvious that the following problem is equivalent to 1.1:
L
l
u
l
: u
l
t
−L
l
u
l
l
u
l
G
l
x, t, u
l
,
u
a
l
,
u
b
l
,
J ∗ u
c
l
,
J ∗ u
d
l
x, t
∈ D
T
,
u
l
Γ
T
0,
a
l
ij
x, t, u
l
u
l
x
j
cos
−→
n,x
i
Γ
T
0,
u
l
h
l
x, t
x, t
∈ S
T
,u
l
x, t
ψ
l
x, t
x, t
∈ Q
l
0
,l 1, ,N.
2.28
We construct two sequences {
u
m
}, {u
m
} from the iteration process
L
l
u
l
m
G
l
x, t,
u
l
m−1
,
u
m−1
a
l
,
u
m−1
b
l
,
J ∗ u
m−1
c
l
,
J ∗ u
m−1
d
l
x, t
∈ D
T
,
L
l
u
l
m
G
l
x, t, u
l
m−1
,
u
m−1
a
l
,
u
m−1
b
l
,
J ∗ u
m−1
c
l
,
J ∗ u
m−1
d
l
x, t
∈ D
T
,
u
l
m
Γ
T
0,
u
l
m
Γ
T
0,
a
l
ij
x, t,
u
l
m
u
l
mx
j
cos
−→
n,x
i
Γ
T
0,
a
l
ij
x, t, u
l
m
u
l
mx
j
cos
−→
n,x
i
Γ
T
0,
u
l
m
h
l
x, t
,u
l
m
h
l
x, t
x, t
∈ S
T
,
u
l
m
x, t
ψ
l
x, t
,u
l
m
x, t
ψ
l
x, t
x, t
∈ Q
l
0
,l 1, ,N, m 1, 2, ,
2.29
where
u
0
u, u
0
u, u
m
u
1
m
, ,u
N
m
,andu
m
u
1
m
, ,u
N
m
.
10 Advances in Difference Equations
Lemma 2.5. The sequences {
u
m
}, {u
m
} given by 2.29 are well defined and possess the regularity
u
m
∈C
β
m
Q
T
, u
mt
∈C
β
m
,β
m
/2
D
T
, u
mx
j
∈C
β
m
,β
m
/2
D
k,T
,
u
mx
i
x
j
∈C
β
m
,β
m
/2
D
k,T
, u
mx
j
t
∈L
2
D
T
for some β
m
∈
0,α
0
,
2.30
and the monotone property
u ≤ u
m−1
≤ u
m
≤ u
m
≤ u
m−1
≤
u
x, t
∈ Q
l
T
,m 1, 2, 2.31
Proof. Let
f
l
m−1
f
l
m−1
x, t
: G
l
x, t,
u
l
m−1
,
u
m−1
a
l
,
u
m−1
b
l
,
J ∗ u
m−1
c
l
,
J ∗ u
m−1
d
l
,
f
l
m−1
f
l
m−1
x, t
: G
l
x, t, u
l
m−1
,
u
m−1
a
l
,
u
m−1
b
l
,
J ∗ u
m−1
c
l
,
J ∗ u
m−1
d
l
.
2.32
Then, for any fixed l, m, l ∈{1, ,N},m∈{1, 2, }, and for given u
m−1
and u
m−1
, problem
2.29 is equivalent to require that u
l
m
u
l
m
ψ
l
for x, t ∈ Q
l
0
, u
l
m
is governed by the
problem for one equation with discontinuous coefficients
L
l
u
l
m
f
l
m−1
x, t
x, t
∈ D
T
,
u
l
m
Γ
T
0,
a
l
ij
x, t,
u
m
u
mx
j
cos
−→
n,x
i
Γ
T
0,
u
l
m
h
l
x, t
x, t
∈ S
T
,
u
l
m
x, 0
ψ
l
x, 0
x ∈ Ω
,
2.33
and u
l
m
is governed by the problem
L
l
u
l
m
f
l
m−1
x, t
x, t
∈ D
T
,
u
l
m
Γ
T
0,
a
l
ij
x, t, u
l
m
u
l
mx
j
cos
−→
n,x
i
Γ
T
0,
u
l
m
h
l
x, t
x, t
∈ S
T
,u
l
m
x, 0
ψ
l
x, 0
x ∈ Ω
.
2.34
Problems 2.33 and 2.34 are the special case of 16, problem 1, 2, 5 for
one equation. Reference 16, Theorem 5 shows that problems 2.33 and 2.34 have a
unique piecewise classical solution
u
l
m
and u
l
m
satisfying 2.30, respectively, whenever
f
l
m−1
x, t,f
l
m−1
x, t ∈C
1β
m−1
D
k,T
k 1, ,K for some β
m−1
∈ 0,α
0
. Furthermore,
Advances in Difference Equations 11
by the formula of integration by parts we get from 2.33 and 2.34 that for, any nonnegative
bounded vector function η η
1
, ,η
N
∈
◦
W
1,1
2
D
T
and any τ ∈ 0,T,
H
l
τ;
u
m
, η
D
τ
l
u
l
m
η
l
dx dt
D
τ
f
l
m−1
x, t
η
l
dx dt,
H
l
τ; u
m
, η
D
τ
l
u
l
m
η
l
dx dt
D
τ
f
l
m−1
x, t
η
l
dx dt.
2.35
We next prove the lemma by the principle of induction. When m 1, Definition 2.2
and hypotheses H-iii and iv show that
u,
u ∈C
α
0
Q
T
∩C
1α
0
D
k,T
,J∗
u,J ∗
u ∈
C
1α
0
D
k,T
and g
l
x, t, u, v ∈ C
1α
0
D
k,T
×S×S
∗
. Thus, for each l 1, ,N, f
l
0
x, t and
f
l
0
x, t are in C
1β
0
D
k,T
for some β
0
∈ 0,α
0
and problems 2.33 and 2.34 for m 1 have
a unique piecewise classical solution
u
l
1
and u
l
1
, respectively. Since the relation
u ≤
u implies
that J ∗
u ≤ J ∗
u, then 2.27 and 2.32 yield that f
l
0
− f
l
0
≤ 0. By using 2.6 and 2.35 for
m 1, we have that
H
l
τ;
u
1
, η
−H
l
τ;
u, η
D
τ
l
u
l
1
−
l
u
l
η
l
dx dt
≤
D
τ
f
l
0
− f
l
0
η
l
dxdt 0,l 1, ,N,
H
l
τ; u
1
, η
−H
l
τ;
u
1
, η
D
τ
l
u
l
1
−
l
u
l
1
η
l
dx dt
D
τ
f
l
0
− f
l
0
η
l
dxdt ≤ 0,l 1, ,N.
2.36
Note that u
1
x, tu
1
x, t ≤
ux, t for x, t ∈ S
T
∪{x, t : x ∈ Ω,t 0}. It follows from part
ii of Lemma 2.4 that u
1
≤ u
1
≤
u for x, t ∈ D
T
. Similar argument gives the relation
u ≤ u
1
for x, t ∈ D
T
. Since u
l
≤ u
l
1
u
l
1
ψ
l
≤ u
l
for x, t ∈ Q
l
0
, the above conclusions show that u
1
and u
1
are well defined and possess the properties 2.30 and 2.31 for m 1.
Assume, by induction, that
u
m
and u
m
given by 2.29 are well defined and possess
the properties 2.30 and 2.31.Thus,
u
l
m
x, tu
l
m
x, tψ
l
x, t for x, t ∈ Q
l
0
.By1.2
and hypothesis H-iv,
J
l
∗ u
l
m
t
0
J
l
x, t − s
u
l
m
x, s
ds
I
l
J
l
x, t − s
ψ
l
x, s
ds ∈ C
1β
∗
m
D
k,T
∩S
∗l
,
J
l
∗ u
l
m
t
0
J
l
x, t − s
u
l
m
x, s
ds
I
l
J
l
x, t − s
ψ
l
x, s
ds ∈ C
1β
∗
m
D
k,T
∩S
∗l
,
J
l
∗ u
l
≤ J
l
∗ u
l
m−1
≤ J
l
∗ u
l
m
≤ J
l
∗ u
l
m
≤ J
l
∗ u
l
m−1
≤ J
l
∗ u
l
,l 1, ,N,
2.37
12 Advances in Difference Equations
where β
∗
m
∈ 0,α
0
. Hypothesis H-iii and 2.37imply that f
l
m
x, t and f
l
m
x, t are in
C
1β
m
D
k,T
k 1, ,K for some β
m
∈ 0,α
0
. Again by using 16, Theorem 5,weobtain
that for each l 1, ,N, problems 2.33 and 2.34 for the case m1 have a unique piecewise
classical solution
u
l
m1
and u
l
m1
, respectively. It follows from 2.27, 2.32, 2.37,and2.35
for the cases m and m 1that
H
l
τ;
u
m1
, η
−H
l
τ;
u
m
, η
D
τ
l
u
l
m1
−
l
u
l
m
η
l
dx dt
D
τ
G
l
x, t,
u
l
m
,
u
m
a
l
,
u
m
b
l
,
J ∗ u
m
c
l
,
J ∗ u
m
d
l
− G
l
x, t,
u
l
m−1
,
u
m−1
a
l
,
u
m−1
b
l
,
J ∗ u
m−1
c
l
,
J ∗ u
m−1
d
l
η
l
dx dt
≤ 0,l 1, ,N,
H
l
τ; u
m1
, η
−H
l
τ;
u
m1
, η
D
τ
l
u
l
m1
−
l
u
l
m1
η
l
dx dt
D
τ
G
l
x, t, u
l
m
,
u
m
a
l
,
u
m
b
l
,
J ∗ u
m
c
l
,
J ∗ u
m
d
l
− G
l
x, t,
u
l
m
,
u
m
a
l
,
u
m
b
l
,
J ∗ u
m
c
l
,
J ∗ u
m
d
l
η
l
dx dt
≤ 0,l 1, ,N.
2.38
Since u
m1
u
m1
u
m
for x, t ∈ S
T
∪{x, t : x ∈ Ω,t 0}, using again part ii of
Lemma 2.4,weobtainthatu
m1
≤ u
m1
≤ u
m
in D
T
. The similar proof gives that u
m
≤ u
m1
in
D
T
.Noticethatu
l
m
u
l
m1
u
l
m1
u
l
m
ψ
l
for x, t ∈ Q
l
0
,l 1, ,N.Wegetthatu
m1
and
u
m1
are well defined and possess the properties 2.30 and 2.31 for the case m 1. By the
principle of induction, we complete the proof of the lemma.
3. Uniform Estimates of {u
m
}, {u
m
}
To prove the existence of solutions to 1.1, in this section, we show the uniform estimates of
{
u
m
}, {u
m
}.
3.1. Preliminaries
In this section we introduce more notations. Let
a
Nl
ij
x, t, v
: a
l
ij
x, t, v
,b
Nl
j
x, t, v
: b
l
j
x, t, v
,h
Nl
x, t
: h
l
x, t
,
J
Nl
x, t
: J
l
x, t
,ψ
Nl
x, t
: ψ
l
x, t
,Q
Nl
0
: Q
l
0
,
ˇ
G
l
x, t, U
m−1
,J ∗ U
m−1
: G
l
x, t,
u
l
m−1
,
u
m−1
a
l
,
u
m−1
b
l
,
J ∗ u
m−1
c
l
,
J ∗ u
m−1
d
l
,
ˇ
G
Nl
x, t, U
m−1
,J ∗ U
m−1
: G
l
x, t, u
l
m−1
,
u
m−1
a
l
,
u
m−1
b
l
,
J ∗ u
m−1
c
l
,
J ∗ u
m−1
d
l
,
U
m
U
1
m
, ,U
2N
m
:
u
m
, u
m
,J∗ U
m−1
:
J
1
∗ U
1
m−1
, ,J
2N
∗ U
2N
m−1
,l 1, ,N.
3.1
Advances in Difference Equations 13
K
ρ
is an arbitrary open ball of radius ρ with center at x
0
,andQ
ρ
is an arbitrary cylinder of
the form K
ρ
× t
0
− ρ
2
, t
0
.K
2ρ
is concentric with K
ρ
. Ω
ρ
: K
ρ
∩ Ω.
In this section, C··· denotes the constant depending only on the parameters
M, a
0
,θ
0
,α
0
,μ
1
,μ
2
,νM,μM,and
0
from hypothesis H and 2.25 and on the
quantities appearing in parentheses, independent of m, where M : max
l1, ,N
{u
l
CQ
l
T
u
l
CQ
l
T
} and
0
: max
1≤l≤N
max
k1, ,K
l
x, t
C
1
D
k,T
.
Write 2.29 in the form
L
l
U
l
m
ˇ
G
l
x, t, U
m−1
,J ∗ U
m−1
x, t
∈ D
T
,
U
l
m
Γ
T
0,
a
l
ij
x, t, U
l
m
U
l
mx
j
cos
−→
n,x
i
Γ
T
0,
U
l
m
h
l
x, t
x, t
∈ S
T
,
U
l
m
x, t
ψ
l
x, t
x, t
∈ Q
l
0
,l 1, ,2N, m 1, 2,
3.2
Consider the equalities
K
k1
T
0
Ω
k
L
l
U
l
m
η
l
dx dt
D
T
ˇ
G
l
x, t, U
m−1
,J∗U
m−1
η
l
dx dt for any
η ηx, tη
1
, ,η
2N
∈
◦
W
1,0
2
D
T
. From the formula of integration by parts, we see that
D
T
a
l
ij
x, t, U
l
U
l
mx
j
η
l
x
i
dxdt
D
T
−U
mt
− b
l
j
x, t, U
l
m
U
l
mx
j
−
l
U
l
ˇ
G
l
x, t, U
m−1
,J ∗ U
m−1
η
l
dx dt,
l 1, ,2N, m 1, 2,
3.3
Similarly, for any φ φxφ
1
, ,φ
2N
∈
◦
W
1
2
Ω and for every t ∈ 0,T,weget
Ω
a
l
ij
x, t, U
l
U
l
mx
j
φ
l
x
i
dx
Ω
−U
mt
− b
l
j
x, t, U
l
m
U
l
mx
j
−
l
U
l
ˇ
G
l
x, t, U
m−1
,J ∗ U
m−1
φ
l
dx,
l 1, ,2N, m 1, 2,
3.4
3.2. Uniform Estimates of U
l
m
C
α
1
,α
1
/2
D
T
, U
l
mx
L
2
D
T
Lemma 3.1. There exist constants α
1
and C depending only on M, a
0
, θ
0
, α
0
, μ
1
, μ
2
, νM, μM,
and
0
, independent of m, such that
U
l
m
C
α
1
,α
1
/2
D
T
≤ C, 0 <α
1
< 1,
3.5
U
l
mx
L
2
D
T
≤ C, l 1, ,2N, m 1, 2,
3.6
14 Advances in Difference Equations
Proof. Fix l, m, l ∈{1, ,2N}, m ∈{1, 2, }.Letw U
l
m
. Then w is the bounded generalized
solution of the following single equation:
L
l
w
ˇ
G
l
x, t, U
m−1
,J ∗ U
m−1
x, t
∈ D
T
3.7
inthesenseof10, Section 1, Chapter V. Equation 3.7 is the special case of 10, Chapter
V, 0.1 with a
i
x, t, w, w
x
a
l
ij
x, t, ww
x
j
and ax, t, w, w
x
b
l
j
x, t, ww
x
j
l
x, tw −
ˇ
G
l
x, t, U
m−1
,J ∗ U
m−1
.From2.31 and hypotheses H-iii–v,weseethat
a
l
i
x, t, u
l
,p
p
i
a
l
ij
x, t, w
p
j
p
i
≥ ν
M
p
2
,
a
i
x, t, w, p
∂a
i
x, t, w, p
∂x
j
∂a
i
x, t, w, p
∂w
≤ C
p
,
a
x, t, w, p
b
l
j
x, t, w
p
j
l
x, t
w −
ˇ
G
l
x, t, U
m−1
,J ∗ U
m−1
≤ C
p
1
,
3.8
where p p
1
, ,p
n
. Then 3.8 and 10, Chapter V, Theorem 1.1 give 3.5, and the proof
similar to that of 10, Chapter V, formula 4.1 gives 3.6.
Lemma 3.2. There exists a positive constant ρ
1
depending only on M,a
0
, θ
0
, α
0
, μ
1
, μ
2
, νM,
μM, and
0
, such that when ρ ≤ ρ
1
, for any cylinder Q
ρ
with x
0
,t
0
∈ D
T
and for any bounded
function ζ ζx, t ∈
◦
W
1, 0
2
Q
ρ
,
Q
ρ
∩D
T
U
l
mx
2
ζ
2
dx dt
≤ Cρ
α
1
Q
ρ
∩D
T
|
ζ
x
|
2
1
U
l
mt
ζ
2
dx dt, l 1, ,2N, m 1, 2,
3.9
and, for any bounded function λ λx ∈
◦
W
1
2
K
ρ
and for every t ∈ 0,T,
Ω
ρ
U
l
mx
2
λ
2
dx ≤ Cρ
α
1
Ω
ρ
|
λ
x
|
2
1
U
l
mt
λ
2
dx, l 1, ,2N, m 1, 2,
3.10
Proof. When K
ρ
⊂ Ω,setη
l
U
l
m
x, t−U
l
m
x
1
,t
1
ζ
2
in 3.3 and φ
l
U
l
m
x, t−U
l
m
x
1
,tλ
2
in 3.4, where x
1
,t
1
is an arbitrary point in Q
ρ
. When K
ρ
∩ ∂Ω
/
∅,setη
l
U
l
m
x, t −
h
l
x, tζ
2
in 3.3 and φ
l
U
l
m
x, t − h
l
x, tλ
2
in 3.4.Thus3.9 and 3.10 follow from
3.8 and the proofs similar to those of 15, formulas 2.7 and 4.2.
3.3. Uniform Estimates on Ω
× 0,T
The bounds in this subsection will be of a local nature. By hypothesis H-i for any given
point x
0
∈ Γ there exists a ball K
ρ
with center at x
0
such that we can straighten Γ ∩ K
ρ
out
introducing new nondegenerate coordinates y yx possessing bounded first and second
Advances in Difference Equations 15
derivatives with respect to x. It is possible to divide Γ into a finite number of pieces and
introduce for each of them coordinates y see 11, Chapter 3, Section 16. Therefore, without
loss of generality we assume that the interface Γ lies i n the plane x
n
0.
In 15, Tan and Leng investigate the H
¨
older estimates for the first derivatives of
the generalized solution u for one parabolic equation with discontinuous coefficients and
without time delays. The estimates u
x
j
C
α
Ω
∩Ω
k
×t
,T
, u
t
C
α
Ω
×t
,T
in 15 depend on
max
t
,T
u
t
L
q/2
Ω
u
t
L
2
Ω
for some q>n, where Ω
⊂⊂ Ω, 0 <t
<T.Theresultsof
15 can not be used directly in this paper, but, by a slight modification, the methods and
the framework of 15 can be used to obtain the uniform estimates of U
l
mx
j
C
α
Ω
∩Ω
k
×0,T
,
U
l
mt
C
α
Ω
×0,T
in this subsection. We omit most of the detailed proofs and only sketch the
main steps. The main changes in the derivations are the following: i15, formulas 2.7 and
4.2 are replaced by 3.9 and 3.10, respectively; ii the estimates in this subsection are
on Ω
× 0,T, while the estimates in 15 are on Ω
× t
,T; iii the behavior of the reaction
functions with continuous delays requires special considerations.
Lemma 3.3. Let K
ρ
,K
2ρ
⊂ Ω. Then there exists a positive constant ρ
2
depending only on M, a
0
, θ
0
,
α
0
, μ
1
, μ
2
, νM, μM, and
0
, such that, when ρ ≤ ρ
2
,
T
0
K
ρ
U
l
mt
2
U
l
mx
4
U
l
mxx
2
dx dt ≤ C
1
ρ
,l 1, ,2N, m 1, 2, ,
3.11
where
T
0
K
ρ
|U
l
mxx
|
2
dx dt :
K
k1
T
0
K
ρ
∩Ω
k
|U
l
mxx
|
2
dx dt.
Proof. Let λ λx, t be an arbitrary smooth function taking values i n 0, 1 such that λ 0
for x/∈ K
2ρ
or t ≤ t
0
− 4ρ
2
,and|λ
x
|
2
|λ
t
|≤C/ρ
2
for x, t ∈ Q
2ρ
. Hypothesis H-iv shows
that for m 1,
T
0
Ω
2ρ
|J
l
∗ U
l
m−1
x
|
2
dx dtl 1, ,2N are estimated by a constant C,and,
for m>1,
T
0
K
2ρ
J
l
∗ U
l
m−1
x
2
dx dt
T
0
K
2ρ
t
0
J
l
x, t − s
U
l
m−1
x, s
ds
I
l
J
l
x, t − s
ψ
l
m−1
x, s
ds
x
2
dx dt
≤ C C
T
0
K
2ρ
U
l
m−1
x
2
dx dt, l 1, ,2N.
3.12
These inequalities, together with 2.11, 3.6,and2.37, imply that
T
0
K
2ρ
d
ˇ
G
l
x, t, U
m−1
,J ∗ U
m−1
dx
s
U
l
mx
s
λ
2
dx dt
≤ C
T
0
K
2ρ
|
U
mx
|
2
J ∗ U
m−1
x
2
dx dt
≤ C, s 1, ,n− 1,l 1, ,2N, m 1, 2,
3.13
16 Advances in Difference Equations
Based on these inequalities, we can get
n−1
s1
T
0
K
2ρ
U
l
mx
s
x
2
λ
2
dx dt ≤ C
1
ρ
T
0
K
2ρ
U
l
mx
4
λ
2
dx dt, l 1, ,2N, m 1, 2,
3.14
For this purpose, similar to 15, Lemma 3.1, we consider not the estimate of the second
derivatives of U
l
but the estimate of the difference ratios Δ/Δx
s
of the first derivatives U
l
x
i
by
setting η
l
Δ/Δx
s
ΔU
l
x−Δx
s
,t/Δx
s
λ
2
x−Δx
s
,t in 3.3, where Δvx, t/Δx
s
,s
1, ,n− 1, denote the difference ratios vx Δx
s
,t − vx, t/Δx
s
with respect to x
s
,and
then we obtain 3.14 by letting Δx
s
→ 0.
We next show that
T
0
K
2ρ
U
l
mt
2
λ
2
dx dt ≤ C
1
ρ
C
T
0
K
2ρ
U
l
mx
4
λ
2
dx dt, l 1, ,2N, m 1, 2,
3.15
To do this, consider
K
k1
T
0
Ω
k
L
l
U
l
m
U
l
mt
λ
l
dx dt
D
T
ˇ
G
l
x, t, U
m−1
,J ∗ U
m−1
U
l
mt
λ
l
dx dt.
From an integration by parts we get
1
2
K
2ρ
a
l
ij
x, t, U
l
m
U
l
mx
j
U
l
mx
i
λ
2
dx
tt
0
t0
T
0
K
2ρ
⎧
⎨
⎩
U
l
mt
2
λ
2
−
1
2
∂a
l
ij
∂U
l
m
U
l
mx
j
U
l
mx
i
U
mt
λ
2
−
1
2
∂a
l
ij
∂t
U
l
mx
j
U
l
mx
i
λ
2
− a
l
ij
U
l
mx
j
U
l
mx
i
λλ
t
2a
l
ij
U
l
mx
j
U
mt
λλ
x
i
b
l
j
U
l
mx
j
l
U
l
m
−
ˇ
G
l
x, t, U
m−1
,J ∗ U
m−1
U
l
mt
λ
2
⎫
⎬
⎭
dx dt 0,
l 1, ,2N, m 1, 2,
3.16
By hypothesis H-iii and Cauchy’s inequality with ε we conclude from the above equalities
that
T
0
K
2ρ
U
l
mt
2
λ
2
dx dt
≤ ε
T
0
K
2ρ
U
l
mt
2
λ
2
dx dt C
C
ε
T
0
K
2ρ
1
U
l
mx
2
λ
2
λ
2
t
|
λ
x
|
2
U
l
mx
4
λ
2
dx dt.
3.17
In view of 3.6, setting ε 1/2, we have 3.15.
Advances in Difference Equations 17
Next, the proof similar to the first inequality of 3.5 of 15 gives that there exists a
positive constant ρ
2
depending only on M, a
0
, θ
0
, α
0
, μ
1
, μ
2
, νM, μM,and
0
, such that,
when ρ ≤ ρ
2
,
T
0
K
2ρ
U
l
mx
4
λ
2
dx dt ≤ C
1
ρ
,l 1, ,2N, m 1, 2, 3.18
Furthermore, since the equations in 3.2 and Hypothesis H-iii show that
U
l
mx
n
x
n
≤ C
U
l
mt
n−1
s1
U
l
mx
s
x
U
l
x
2
1
x, t
∈ D
k,T
,k 1, ,K, 3.19
then 3.11 follows from 3.14–3.19.
Lemma 3.4. Let K
ρ
,K
2ρ
⊂ Ω. Then there exists a positive constant ρ
3
depending only on M, a
0
,
θ
0
,α
0
, μ
1
, μ
2
, νM, μM, and
0
, such that, when ρ ≤ ρ
3
,
max
0,T
K
2ρ
|
U
mt
|
r1
dx
T
0
K
2ρ
|
U
mt
|
r−1
|
U
mtx
|
2
|
U
mt
|
r2
dx dt
≤ C
q,
1
ρ
.r 1, ,q, m 1, 2, ,
3.20
where |U
mt
| :
2N
l1
|U
l
mt
|
2
1/2
, |U
mtx
| :
2N
l1
|U
l
mtx
|
2
1/2
.
Proof. Let λ λx, t be an arbitrary smooth function taking values in 0, 1 such that λ 0
for x/∈ K
2ρ
or t ≤ t
0
− 4ρ
2
,and|λ
x
|
2
|λ
t
|≤C/ρ
2
for x, t ∈ Q
2ρ
. Similar to 3.12,from
hypotheses H-iii-iv, 2.37 for the case m − 1, and H
¨
older’s inequality we see that
τ
0
K
2ρ
d
ˇ
G
l
x, t, U
m−1
,J ∗ U
m−1
dt
U
l
mt
r
λ
2
dx dt
≤ C
τ
0
K
2ρ
|
U
mt
|
r1
U
m−1t
r1
dx dt, l 1, ,2N, m 1, 2,
3.21
Next, let us examine the difference ratio with respect to t on both sides of L
l
U
l
m
ˇ
G
l
x, t, U
m−1
,J∗U
m−1
. Multiplying the equations obtained by |U
l
mt
|
r−1
U
l
mt
λ
2
, where U
l
mt
U
l
m
x, tΔt − U
l
m
x, t/Δt, integrating by parts, and then letting Δt → 0, from 3.9 and
the proof similar to that of 15, formula 3.26, we find that there exists a positive constant
ρ
3,1
depending only on M, a
0
, θ
0
, α
0
, μ
1
, μ
2
, νM, μM,and
0
, such that, when ρ ≤ ρ
3,1
,
18 Advances in Difference Equations
K
2ρ
|
U
mt
|
r1
λ
2
dx
tτ
t0
τ
0
K
2ρ
|
U
mt
|
r−1
|
U
mtx
|
2
λ
2
dx dt ≤ Cρ
α
1
τ
0
K
2ρ
|
U
mt
|
r2
λ
2
dx dt
C
τ
0
K
2ρ
1
|
U
mt
|
r1
λ
2
λ
|
λ
t
|
|
λ
x
|
2
1
|
U
m−1
t
|
r1
λ
2
dx dt.
3.22
To estimate
τ
0
K
2ρ
|U
mt
|
r2
λ
2
dxdt, we take η |U
l
mt
|
r
U
l
mt
λ
2
in 3.3. Hence, by
hypotheses H-iii-iv and Cauchy’s inequality we get
τ
0
K
2ρ
|
U
mt
|
r2
λ
2
dx dt
≤ C
q
τ
0
K
2ρ
|
U
mt
|
r−1
|
U
mtx
|
2
λ
2
|
U
mx
|
2
|
U
mt
|
r1
ξ
2
|
U
mt
|
r1
λ
2
|
λ
x
|
2
dx dt,
3.23
and by 3.9 with ζ |U
mt
|
2
r1/4
λ we get
τ
0
K
2ρ
|
U
mx
|
2
|
U
mt
|
r1
λ
2
dx dt
≤ C
q
ρ
α
1
τ
0
K
2ρ
|
U
mt
|
r−1
|
U
mtx
|
2
λ
2
1
|
U
mt
|
r1
|
λ
x
|
2
|
U
mt
|
r2
λ
2
dx dt.
3.24
Furthermore, 3.22–3.24 show that
τ
0
K
2ρ
|
U
mt
|
r2
λ
2
dx dt ≤ C
3,1
q
ρ
α
1
τ
0
K
2ρ
|
U
mt
|
r2
λ
2
dx dt
C
τ
0
K
2ρ
1
|
U
mt
|
r1
λ
2
λ
|
λ
t
|
|
λ
x
|
2
1
|
U
m−1
t
|
r1
λ
2
dx dt.
3.25
Set ρ
3,2
: min{ρ
3,1
, 2C
3,1
q
−1/α
1
}. Thus, when 0 <ρ≤ ρ
3,2
,
τ
0
K
2ρ
|
U
mt
|
r2
λ
2
dx dt
≤ C
τ
0
K
2ρ
1
|
U
mt
|
r1
λ
2
λ
|
λ
t
|
|
λ
x
|
2
1
|
U
m−1
t
|
r1
λ
2
dx dt.
3.26
Note that by property 2.30, hypothesis H-iii, and the equations in 3.2,
U
l
mt
x, 0
d
dx
i
a
l
ij
x, 0,ψ
l
ψ
l
x
j
b
l
j
x, t, ψ
l
ψ
l
x
j
ˇ
G
l
x, 0, ψ,J ∗ ψ
x ∈
Ω
,
3.27
Advances in Difference Equations 19
where ψ :ψ
1
, ,ψ
2N
. Therefore,
Ω
|U
mt
x, 0|
r
dx can be estimated from above by Cq.
Thus, using the same arguments given in the derivation of 15, f ormula 3.29,weget3.20
from 3.26, 3.22 ,and3.27.
Lemma 3.5. Let K
ρ
,K
2ρ
⊂ Ω. For any given positive integer q, one has that
K
ρ
U
l
mxx
2
1
U
l
mx
2r
U
l
mx
2r4
dx
≤ C
q,
1
ρ
,r 0, 1, ,q, l 1, ,2N, m 1, 2,
3.28
for every t ∈ 0,T.
Proof. By using 3.20 and 10, Chapter II, Lemmas 5.2 and 5.3
, from the same argument as
that in the proof of 15, formula 2.2, we find that, f or every t ∈ 0,T,
K
ρ
U
l
mt
2
ζ
2
dx ≤ Cρ
α
1
K
ρ
|
ζ
x
|
2
dx, l 1, ,2N, m 1, 2, , 3.29
where ζ ζx is an arbitrary bounded function from
◦
W
1
2
K
ρ
. Then by 3.4, 3.8, 3.10,
and 3.29, the proof similar to 15, formula 4.6 implies 3.6.
Based on the above uniform estimates, we can get the following local H
¨
older estimates
of the first derivatives.
Lemma 3.6. Let K
ρ
⊂ Ω
⊂⊂ Ω. There exist positive constants α
2
, α
3
, and Cd
, 0 <α
2
,α
3
< 1,
such that
max
Q
ρ
∩D
k,T
U
l
x
j
ρ
−α
2
osc
U
l
x
j
,Q
ρ
∩ D
k,T
≤ C
d
,j 1, ,n, k 1, ,K, l 1, ,2N,
3.30
max
Q
ρ
∩D
T
U
l
t
ρ
−α
3
osc
U
l
t
,Q
ρ
∩ D
T
≤ C
d
,l 1, ,2N, m 1, 2, ,
3.31
where α
2
and α
3
depend only on d
and the parameters M, a
0
, θ
0
, α
0
, μ
1
, μ
2
, νM, μM, and
0
,
independent of m.
Proof. By Hypothesis H, 3.20,and3.27, the proof similar to that of 15, Lemma 4.4 gives
3.31,and,by3.20 and 3.28, the proof similar to that of 15, Lemma 4.3 gives
max
K
ρ
U
l
x
s
ρ
−β
∗
1
osc
U
l
x
s
,K
ρ
≤ C
d
,s 1, ,n− 1,l 1, ,2N,
3.32
max
K
ρ
∩Ω
k
U
l
x
n
ρ
−β
∗
2
osc
U
l
x
n
,K
ρ
∩ Ω
k
≤ C
d
,l 1, ,2N, m 1, 2, , k 1, ,K,
3.33
20 Advances in Difference Equations
where β
∗
1
and β
∗
2
depend only on d
and the parameters M, a
0
, θ
0
, α
0
, μ
1
, μ
2
, νM, μM,and
0
.Byusing3.5, 3.32, 3.33,and10, Chapter II, Lemma 3.1 we see that for any given
k k 1, ,K,
U
l
mx
j
x, t
1
− U
l
mx
j
x, t
2
≤ C
d
|
t
1
− t
2
|
β
∗
3
x, t
1
,
x, t
2
∈
Ω
∩ Ω
k
×
0,T
,j 1, ,n,
3.34
where β
∗
3
α
1
/2 minβ
∗
1
,β
∗
2
/1 minβ
∗
1
,β
∗
2
. Then 3.30 follows from 3.32–3.34.
3.4. Uniform Estimates on D
T
Theorem 3.7. Let hypothesis (H) holds, and let the sequence {U
m
} be given by 3.2.Then
U
l
mx
j
C
α
4
,α
4
/2
D
k,T
U
l
mt
C
α
4
,α
4
/2
D
T
≤ C, 0 <α
4
< 1,
3.35
U
l
mx
i
x
j
L
2
D
k,T
U
l
mx
j
t
L
2
D
T
≤ C, k 1, ,K, l 1, ,2N,
3.36
where α
4
depends only on M, a
0
, θ
0
, α
0
, μ
1
, μ
2
, νM, μM, and
0
, independent of m. For any
given k, k ∈{1, ,K}, letting Ω
⊂⊂ Ω
k
and t
<T, there exists a positive constant α
5
∈ 0, 1
depending only on d
: distΩ
,∂Ω
k
,t
and the parameters M,a
0
, θ
0
,α
0
, μ
1
, μ
2
, νM, μM,
and
0
, such that
U
l
m
C
2α
5
,1α
5
/2
Ω
×t
,T
≤ C
d
,t
,l 1, ,2N, m 1, 2, 3.37
Proof. Since Γ∩∂Ω∅, then there exists a subdomain of Ω, denoted by Ω
K
, such that ∂Ω ⊂ Ω
K
.
Then the coefficients of the equations L
l
U
l
m
ˇ
G
l
x, t, U
m−1
,J ∗ U
m−1
are continuous in Ω
K
.
In 10, the estimates near ∂Ω for the equations with continuous coefficients and without
time delays are well known. By the methods of Section 3.3 and 10 we can get the estimates
near ∂Ω. The details are omitted. Then the estimates near ∂Ω and the results of the above
subsections give 3.35 and 3.36 .
We next prove 3.37. For any fixed l, m, k, l ∈{1, ,2N}, m ∈{1, 2, }, k ∈
{1, ,K},U
l
m
satisfies the linear equation with continuous coefficients
U
l
mt
− a
ij
x, t
U
l
mx
i
x
j
b
j
x, t
U
l
mx
j
f
x, t
x, t
∈ Ω
k
×
0,T
,
3.38
where
a
ij
x, t
a
l
ij
x, t, U
l
m
,
b
j
x, t
−
∂a
l
ij
x, t, U
l
m
∂U
l
m
U
l
mx
i
−
∂a
l
ij
x, t, U
l
m
∂x
i
b
l
j
x, t, U
l
m
,
f
x, t
−
l
x, t
U
l
m
ˇ
G
l
x, t, U
m−1
,J ∗ U
m−1
.
3.39
Advances in Difference Equations 21
It follows from 3.35, 3.36, and hypotheses H-iii-iv that
a
ij
x, t
;
b
j
x, t
;
f
x, t
C
β
∗
4
D
k,T
≤ C, i, j 1, ,n,
3.40
where β
∗
4
∈ 0, 1 depends only on α
4
and the parameters M, a
0
,θ
0
,α
0
,μ
1
, μ
2
, νM, μM,and
0
. Therefore, 3.40 and the Schauder estimate for linear parabolic equation yield 3.37.
4. Existence and Uniqueness of Solutions for 1.1
In this section we show that the sequences {u
m
}, {u
m
} converge to the unique solution of
1.1 and prove the main theorem of this paper.
Theorem 4.1. Let hypothesis (H) hold. Then, problem 1.1 has a unique piecewise classical solution
u
∗
in S, and the sequences {u
m
}, {u
m
} given by 2.29 converge monotonically to u
∗
. The relation
u ≤ u
m−1
≤ u
m
≤ u
∗
≤ u
m
≤ u
m−1
≤
u
x, t
∈ Q
l
T
,m 1, 2, 4.1
holds.
Proof. It follows from Lemma 2.5 that the pointwise limits
lim
m →∞
u
m
u, lim
m →∞
u
m
u
4.2
exist and satisfy the relation
u ≤ u
m−1
≤ u
m
≤ u ≤ u ≤ u
m
≤ u
m−1
≤
u. 4.3
Let {u
m
} denote either the sequence {u
m
} or the sequence {u
m
},andletu be the
corresponding limit.
Estimates 3.5, 3.6, 3.35,and3.36 imply that there exists a subsequence {u
m
}
denoted by {u
m
} still such that {u
m
} and {u
mt
} converge in CD
T
to u and u
t
, respectively,
for each i, j 1, ,n, {u
mx
j
} converges in CD
k,T
to u
x
j
, {u
mx
i
x
j
} converges weakly in
L
2
D
k,T
to u
x
i
x
j
,and{u
mx
j
t
} converges weakly in L
2
D
T
to u
x
j
t
k 1, ,K.Thus,
u ∈C
α
1
,α
1
/2
D
T
, u
x
j
∈C
α
4
,α
4
/2
D
k,T
, u
t
∈C
α
4
,α
4
/2
D
T
,andu
x
j
t
∈L
2
D
T
. Since u
l
ψ
l
in Q
l
0
,l 1, ,N, then u ∈C
α
1
,α
1
/2
Q
T
. For any given k, k 1, ,K, and any given
Ω
⊂⊂ Ω
k
and t
∈ 0,T, 3.37 in Theorem 3.7 implies that there exists a subsequence
{u
m
} denoted by {u
m
} still such that {u
m
} converges in C
2,1
Ω
× t
,T to u. Then
u ∈C
2α
5
,1α
5
/2
Ω
× t
,T.
22 Advances in Difference Equations
Let m →∞. The above conclusions and 2.29 yield that
u, u satisfy
u
l
t
−L
l
u
l
g
l
x, t,
u
l
,
u
a
l
,
u
b
l
,
J ∗ u
c
l
,
J ∗ u
d
l
x, t
∈ D
T
,
u
l
t
−L
l
u
l
g
l
x, t, u
l
,
u
a
l
,
u
b
l
,
J ∗ u
c
l
,
J ∗ u
d
l
x, t
∈ D
T
,
u
l
Γ
T
0,
u
l
Γ
T
0,
a
l
ij
x, t,
u
l
u
l
x
j
cos
−→
n,x
i
Γ
T
0,
a
l
ij
x, t, u
l
u
l
x
j
cos
−→
n,x
i
Γ
T
0,
u
l
h
l
x, t
,u
l
h
l
x, t
x, t
∈ S
T
,
u
l
x, t
ψ
l
x, t
,u
l
x, t
ψ
l
x, t
x, t
∈ Q
l
0
,l 1, ,N, m 1, 2,
4.4
Furthermore, 2.35 shows that
u, u satisfy 2.16 with u, v,q
l
and the symbol “≤” replaced
by u
, u,g
l
, and the symbol “”, respectively. By Lemma 2.4 we get that u u for x, t ∈ D
T
.
In view of
u
l
x, tu
l
x, tψ
l
x, t for x, t ∈ Q
l
0
, then u
l
u
l
for x, t ∈ Q
l
T
,l 1, ,N.
Consequently, by 4.4 and Definition 2.3, u
∗
: u u is a piecewise classical solution of 1.1
in S and satisfies the relation 4.1.Ifu
∗∗
is also a piecewise classical solution of 1.1 in S,
then by Lemma 2.4 the same argument shows that u
∗
≡ u
∗∗
. Therefore, the piecewise classical
solution of 1.1 in S is unique.
Since T is an arbitrary positive number, the piecewise classical solution u
∗
given by
Theorem 4.1 is global.
5. Applications in Ecology
Consider 2-species Volterra-Lotka models with diffusion and continuous delays see 2, 3.
Suppose that the natural conditions for the subdomains Ω
k
,k 1, ,K,aredifferent. Then
the diffusion coefficients are allowed to be discontinuous on the interface Γ. Assume that near
Γ, the density and the flux are continuous. Then
u
l
Γ
T
0,
a
l
ij
x, t, u
l
u
l
x
j
cos
−→
n,x
i
Γ
T
0,l 1, 2,
5.1
where u
1
,u
2
are the densities of the populations of the two species. Therefore, u
1
,u
2
are
governed by the system 1.1, where the reaction functions are explicitly given as follows.
1 For the Volterra-Lotka cooperation model with continuous delays,
g
1
x, t, u,J ∗ u
u
1
r
1
k
− δ
1
k
u
1
σ
1
k
J
2
∗ u
2
x, t
∈ D
k,T
,k 1, ,K,
g
2
x, t, u,J ∗ u
u
2
r
2
k
δ
2
k
J
1
∗ u
1
− σ
2
k
u
2
x, t
∈ D
k,T
,k 1, ,K.
5.2
Advances in Difference Equations 23
2 For the Volterra-Lotka competition model with continuous delays,
g
1
x, t, u,J ∗ u
u
1
r
1
k
− δ
1
k
u
1
− σ
1
k
J
2
∗ u
2
x, t
∈ D
k,T
,k 1, ,K,
g
2
x, t, u,J ∗ u
u
2
r
2
k
− δ
2
k
J
1
∗ u
1
− σ
2
k
u
2
x, t
∈ D
k,T
,k 1, ,K.
5.3
3 For the Volterra-Lotka prey-predator model with continuous delays,
g
1
x, t, u,J ∗ u
u
1
r
1
k
− δ
1
k
u
1
− σ
1
k
J
2
∗ u
2
x, t
∈ D
k,T
,k 1, ,K,
g
2
x, t, u,J ∗ u
u
2
r
2
k
δ
2
k
J
1
∗ u
1
− σ
2
k
u
2
x, t
∈ D
k,T
,k 1, ,K.
5.4
Here r
l
k
, δ
l
k
,andσ
l
k
are all positive constants for k 1, ,K, l 1, 2.
Theorem 5.1. Let the functions a
l
ij
x, t, u
l
, b
l
j
x, t, u
l
, h
l
x, t, and ψ
l
x, t,l 1, 2, satisfy
the hypotheses in (H). If h
l
x, t and ψ
l
x, t, l 1, 2, are nonnegative functions and the
condition
b
2
/b
1
< c
2
/c
1
holds for the cooperation model, where b
1
min
k1, ,K
δ
1
k
/r
1
k
, c
1
max
k1, ,K
σ
1
k
/r
1
k
, b
2
max
k1, ,K
δ
2
k
/r
2
k
, and c
2
min
k1, ,K
σ
2
k
/r
2
k
, and if N 2 and
g
l
x, t, u,J∗u, l 1, 2, are given by one of 5.2–5.4, then problem 1.1 has a unique nonnegative
piecewise classical solution.
Proof. By Theorem 4.1, the proof of this theorem is completed if there exist a pair of coupled
weak upper and lower solutions
u M
1
,M
2
,
u 0, 0 for each case of 5.2–5.4, where
M
1
and M
2
are positive constants. We next prove the existence of M
1
and M
2
for each case.
Note that 1.2 and 2.13 imply that J ∗ M
1
,M
2
M
1
,M
2
.
Case 1. g
l
x, t, u,J ∗ u, l 1, 2, are given by 5.2. Then g·, u, vg
1
·, u, v,g
2
·, u, v is
quasimonotone nondecreasing. The requirement of
u M
1
,M
2
,
u 0, 0 in Definition 2.2
becomes
M
1
1 −
δ
1
k
r
1
k
M
1
σ
1
k
r
1
k
M
2
≤ 0,M
2
1
δ
2
k
r
2
k
M
1
−
σ
2
k
r
2
k
M
2
≤ 0,k 1, ,K. 5.5
Since
b
2
/b
1
< c
2
/c
1
, by the argument in 1, Page 676 we conclude that there exist positive
constants η
1
and η
2
such that, for any R ≥ 1,
1 −
b
1
Rη
1
c
1
Rη
2
≤ 0, 1 b
2
Rη
1
− c
2
Rη
2
≤ 0.
5.6
There exists R
0
such that R
0
η
l
≥ h
l
x, t for x, t ∈ S
T
and R
0
η
l
≥ ψ
l
x, t for x, t ∈ Q
l
0
,l
1, 2. If M
1
,M
2
≥ R
0
η
1
,R
0
η
2
, then M
1
,M
2
satisfies 5.5,and
u M
1
,M
2
,
u 0, 0
are a pair of coupled weak upper and lower solutions of 1.1.
24 Advances in Difference Equations
Case 2. g
l
x, t, u,J ∗ u,l 1, 2, are given by 5.3. g·, u, v is mixed quasimonotone. The
requirement of
u M
1
,M
2
,
u 0, 0 in Definition 2.2 becomes
M
1
r
1
k
− δ
1
k
M
1
≤ 0,M
2
r
2
k
− σ
2
k
M
2
≤ 0,k 1, ,K. 5.7
If M
1
,M
2
≥ max
k1, ,K
r
1
k
/δ
1
k
, max
k1, ,K
r
2
k
/σ
2
k
, M
l
≥ h
l
x, t for x, t ∈ S
T
,andM
l
≥
ψ
l
x, t for x, t ∈ Q
l
0
, l 1, 2, then
u M
1
,M
2
,
u 0, 0 are a pair of coupled weak upper
and lower solutions of 1.1.
Case 3. g
l
x, t, u,J ∗ u, l 1, 2, are given by 5.4. g·, u, v is mixed quasimonotone. The
requirement of
u M
1
,M
2
,
u 0, 0 in Definition 2.2 becomes
M
1
r
1
k
− δ
1
k
M
1
≤ 0,M
2
r
2
k
δ
2
k
M
1
− σ
2
k
M
2
≤ 0,k 1, ,K. 5.8
We first choose M
1
satisfying M
1
≥ max
k1, ,K
r
1
k
,/δ
1
k
,M
1
≥ h
1
x, t for x, t ∈ S
T
and
M
1
≥ ψ
1
x, t for x, t ∈ Q
1
0
, and then we choose M
2
satisfying M
2
≥ max
k1, ,K
r
2
k
/σ
2
k
δ
2
k
M
1
/σ
2
k
, M
2
≥ h
2
x, t for x, t ∈ S
T
,andM
2
≥ ψ
2
x, t for x, t ∈ Q
2
0
.Thus,
u
M
1
,M
2
,
u 0, 0 are a pair of coupled weak upper and lower solutions of 1.1.
Acknowledgments
The author 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 Sichuan College of Education
CJYKT09-024.
References
1 C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, NY, USA, 1992.
2 C. V. Pao, “Systems of parabolic equations with continuous and discrete delays,” Journal of
Mathematical Analysis and Applications, vol. 205, no. 1, pp. 157–185, 1997.
3 C. V. Pao, “Convergence of solutions of reaction-diffusion systems with time delays,” Nonlinear
Analysis: Theory, Methods & Applications, vol. 48, no. 3, pp. 349–362, 2002.
4 C. V. Pao, “Coupled nonlinear parabolic systems with time delays,” Journal of Mathematical Analysis
and Applications, vol. 196, no. 1, pp. 237–265, 1995.
5 J. Liang, H Y. Wang, and T J. Xiao, “On a comparison principle for delay coupled systems with
nonlocal and nonlinear boundary conditions,” Nonlinear Analysis: Theory, Methods & Applications, vol.
71, no. 12, pp. e359–e365, 2009.
6 Y. Chen and M. Wang, “Asymptotic behavior of solutions of a three-species predator-prey model with
diffusion and time delays,” Applied Mathematics Letters, vol. 17, no. 12, pp. 1403–1408, 2004.
7 Y. Li and J. Wu, “Convergence of solutions for Volterra-Lotka prey-predator systems with time
delays,” Applied Mathematics Letters, vol. 22, no. 2, pp. 170–174, 2009.
8 Y M. Wang, “Asymptotic behavior of solutions for a class of predator-prey reaction-diffusion systems
with time delays,” Journal of Mathematical Analysis and Applications, vol. 328, no. 1, pp. 137–150, 2007.
9 L. I. Kamynin, “The method of heat potentials for a parabolic equation withf discontinuous
coefficients,” Sibirski
˘
ı Matemati
ˇ
ceski
˘
ı
ˇ
Zurnal, vol. 4, pp. 1071–1105, 1963 Russian.
10 O. A. Ladyzenskaya, V. A. Solonnikov, and N. N. Ural’ceva, Linear and Quasilinear Equations of Parabolic
Type, American Mathematical Society, Providence, RI, USA, 1968.
Advances in Difference Equations 25
11 O. A. Ladyzhenskaya and N. N. Ural’ceva, Linear and Quasilinear Elliptic Equations, Academic Press,
New York, NY, USA, 1968.
12 O. A. Lady
ˇ
zenskaja, V. Ja. Rivkind, and N. N. Ural’ceva, “Solvability of diffraction problems in
the classical sense,” Trudy Matematicheskogo Instituta imeni V.A. Steklova, vol. 92, pp. 116–146, 1966
Russian.
13 M. Xing, “Regularity of solutions to second-order elliptic equations with discontinuous coefficients,”
Acta Mathematica Scientia Series A, vol. 25, no. 5, pp. 685–693, 2005 Chinese.
14 Q. J. Tan and Z. J. Leng, “The regularity of the weak solutions for the n-dimensional quasilinear
elliptic equations with discontinuous coefficients,” Journal of Mathematical Research & Exposition, vol.
28, no. 4, pp. 779–788, 2008.
15 Q J. Tan and Z J. Leng, “The regularity of generalized solutions for the n-dimensional quasi-linear
parabolic diffraction problems,” Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no. 12, pp.
4624–4642, 2008.
16 G. Boyadjiev and N. Kutev, “Diffraction problems for quasilinear reaction-diffusion systems,”
Nonlinear Analysis: Theory, Methods & Applications, vol. 55, no. 7-8, pp. 905–926, 2003.
