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
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
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
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
,
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
∂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
≥−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
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
: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
,
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
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
◦
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
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
. 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
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
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
u
b
l
,
J ∗
u
c
l
,
J ∗
u
d
l
η
l
dx dt,
H
l
τ;
u, η
c
l
,
J ∗
u
d
l
η
l
dx dt,
u
l
≤ g
l
x, t
≤ u
l
x, t
∈ S
T
,
,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
.
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
, 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
u
l
n
i
1
ξ
2
i
,
2.9
a
l
ij
a
l
ji
,
g
l
x, t, u, v
C
1
D
k,T
×S×S
∗
≤ μ
1
, 2.11
h
l
C
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
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
I
l
J
l
x, t − s
ψ
l
x, s
ds; J
l
∗ u
l
; J
l
∗ u
l
C
,
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
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
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
l
η
l
dx dt
≤H
l
τ; u, η
−
D
τ
q
l
x, t, u
l
,
u
a
l
,
v
∈ 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
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
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
≤
N
l1
D
τ
q
l
x, t, v
l
,
v
a
l
,
u
b
l
,
J ∗ v
c
l
,
J ∗ v
d
l
w
l
dx dt
N
l1
D
τ
⎡
⎣
E
l
1l
w
l
J
l
∗w
l
∈
J∗w
c
l
E
l
2l
J
l
∗ w
l
J
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
a
l
,
Y
θ
b
l
,
Z
θ
c
l
,
Z
θ
d
l
: θ
v
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
≤−E
l
1l
w
l
for w
l
∈
w
b
l
,
E
l
2l
J
l
for J
l
∗ w
l
∈
J ∗ w
c
l
,
−E
l
2l
J
l
∗ w
l
≤−E
l
2l
∈
J ∗ w
d
l
,
2.20
E
l
1l
E
l
2l
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
2
ds
t
0
w
l
x, s
2
ds
dx dt
≤ C
D
τ
τ
0
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
l
ij
x, t, v
l
− a
l
ij
x, t, u
l
u
l
x
j
w
l
x
i
b
l
Ω
w
x, τ
2
dx
ν
O
− ε
N
l1
D
τ
w
|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
dx dt ≤ C
O, O
1
D
τ
|
w
|
2
|
J ∗ w
|
2
dx dt
≤ C
O, O
1
D
τ
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
g
l
x, t, u
l
,
u
a
l
,
u
b
l
,
v
c
l
,
v
d
l
l
,
u
∗
b
l
,
v
c
l
,
v
∗
d
l
≤ G
l
·,u
∗l
,
u
l
: u
l
t
−L
l
u
l
l
u
l
G
l
x, t, u
l
,
u
a
l
,
u
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
l
u
l
m
G
l
x, t,
u
l
m−1
,
u
m−1
a
l
,
u
m−1
b
l
,
J ∗ 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
l
m
u
l
mx
j
cos
−→
n,x
i
Γ
T
0,
a
l
ij
x, t, u
l
m
u
l
mx
T
,
u
l
m
x, t
ψ
l
x, t
,u
l
m
x, t
ψ
l
x, t
x, t
∈ Q
.
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
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
a
l
,
u
m−1
b
l
,
J ∗ u
m−1
c
l
,
J ∗ u
m−1
d
l
,
f
l
m−1
f
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
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
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
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
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
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
, 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;
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
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
∗ 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
I
l
J
l
x, t − s
ψ
l
x, s
ds ∈ C
1β
∗
m
D
k,T
∩S
∗l
,
J
l
∗
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
l
u
l
m
η
l
dx dt
D
τ
G
l
x, t,
u
l
m
,
u
m
a
l
,
u
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
η
l
dx dt
D
τ
G
l
x, t, u
l
m
,
u
m
a
l
,
u
m
b
l
,
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
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
,
J
Nl
x, t
: J
l
x, t
,ψ
Nl
x, t
: ψ
l
x, t
,Q
Nl
0
: Q
l
0
,
ˇ
G
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
,
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, 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
,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
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
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
∈
◦
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
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
< 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
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
∂a
i
x, t, w, p
∂x
j
∂a
i
x, t, w, p
∂w
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
W
1, 0
2
Q
ρ
,
Q
ρ
∩D
T
U
l
mx
2
ζ
2
dx dt
≤ Cρ
α
1
Q
ρ
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
|
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
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
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
U
l
mt
2
U
l
mx
4
U
l
mxx
2
dx dt ≤ C
|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
x
2
dx dt
T
0
K
2ρ
t
0
J
l
x, t − s
U
l
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
λ
2
dx dt
≤ C
T
0
K
2ρ
|
U
mx
|
2
J ∗ U
m−1
x
2
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
s
→ 0.
We next show that
T
0
K
2ρ
U
l
mt
2
λ
2
dx dt ≤ C
1
ρ
C
T
0
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
K
2ρ
⎧
⎨
⎩
U
l
mt
2
λ
2
−
1
2
∂a
l
ij
∂U
l
m
U
l
mx
j
U
l
mx
i
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
U
l
mt
2
λ
2
dx dt
≤ ε
T
0
K
2ρ
U
l
mt
2
λ
2
dx dt C
C
ε
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
,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
,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
|
≤ 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
K
2ρ
d
ˇ
G
l
x, t, U
m−1
,J ∗ U
m−1
dt
U
l
mt
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
r1
λ
2
dx
tτ
t0
τ
0
K
2ρ
|
U
mt
|
r−1
|
U
mtx
|
2
λ
2
dx dt ≤ Cρ
α
λ
2
λ
|
λ
t
|
|
λ
x
|
2
1
|
U
m−1
t
|
r1
λ
2
dx dt.
K
2ρ
|
U
mt
|
r2
λ
2
dx dt
≤ C
q
τ
0
K
2ρ
|
U
mt
|
r−1
|
U
mtx
|
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
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
0
K
2ρ
1
|
U
mt
|
r1
λ
2
λ
|
λ
t
|
|
λ
x
|
2
1
|
2ρ
|
U
mt
|
r2
λ
2
dx dt
≤ C
τ
0
K
2ρ
1
|
U
mt
|
r1
λ
2
λ
|
λ
t
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
,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
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
¨
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
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
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
j
x, t
1
− U
l
mx
j
x, t
2
≤ C
d
|
t
1
− t
2
|
β
∗
3
∗
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
U
l
mx
i
x
j
L
2
D
k,T
U
l
mx
j
t
L
2
: 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
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
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
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
∈ 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
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
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
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
x, t,
u
l
,
u
a
l
,
u
b
l
,
J ∗ u
c
l
,
J ∗ u
d
l
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
,u
l
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
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
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
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
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
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
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
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
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
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
, 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
,
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
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.
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