# A Reaction-Diffusion System with Nonlinear Nonlocal Boundary Conditions

Authors: Alexander Gladkov; Alexandr Nikitin
Journal: International Journal of Partial Differential Equations
 (2014)

---

## Abstract

We consider initial boundary value problem for a reaction-diffusion system with nonlinear and nonlocal boundary conditions and nonnegative initial data. We prove local existence, uniqueness, and nonuniqueness of solutions.


---

## Body

## 1. Introduction

In this paper we consider the following semilinear reaction-diffusion system with nonlinear nonlocal boundary conditions:(1)
u
t
=
Δ
u
+
c
1
(
x
,
t
)
v
p
,
v
t
=
Δ
v
+
c
2
(
x
,
t
)
u
q
,
x
∈
Ω
,
0
<
t
<
T
,
u
(
x
,
t
)
=
∫
Ω
k
1
(
x
,
y
,
t
)
u
m
(
y
,
t
)
d
y
,
x
∈
∂
Ω
,
0
<
t
<
T
,
v
(
x
,
t
)
=
∫
Ω
k
2
(
x
,
y
,
t
)
v
n
(
y
,
t
)
d
y
,
x
∈
∂
Ω
,
0
<
t
<
T
,
u
(
x
,
0
)
=
u
0
(
x
)
,
v
(
x
,
0
)
=
v
0
(
x
)
,
x
∈
Ω
,
where p
,
q
,
m
,
n
>
0, Ω is a bounded domain in ℝ
N  for  N
≥
1 with smooth boundary ∂
Ω
,
T
>
0. Here, c
1
(
x
,
t
)
,
c
2
(
x
,
t
) are nonnegative Hölder continuous functions defined for x
∈
Ω
¯, and t
∈
[
0
,
T
] and k
1
(
x
,
y
,
t
)
,
k
2
(
x
,
y
,
t
) are nonnegative continuous functions defined for x
∈
∂
Ω, y
∈
Ω
¯, and t
∈
[
0
,
T
]. The initial data u
0
(
x
)
,
v
0
(
x
) are nonnegative continuous functions satisfying the boundary conditions at t
=
0.In the past several decades, many physical phenomena have been formulated into nonlocal mathematical models. Initial boundary value problem for semilinear reaction-diffusion equations and systems with nonlocal boundary conditions has been analyzed by many authors (see, e.g., [1–12] and the references therein). Local and global existence, comparison principle, and various qualitative properties have been discussed.We note that formax
⁡
⁡
(
p
,
q
,
m
,
n
)
<
1 the nonlinearities in (1) are non-Lipschitzian. The problem of uniqueness and nonuniqueness for different nonlinear parabolic equations and systems with non-Lipschitzian data has been addressed by several authors. See, for example, [13–16] for equations and [17–21] for systems. In particular, the authors of [16] have investigated the uniqueness of solution for a problem (1) with a single equation.In [7] the authors have considered a problem (1) with c
i
(
x
,
t
)
=
1 and k
i
(
x
,
y
,
t
)
=
k
i
(
x
,
y
)
,
i
=
1,2. They have proved a comparison principal and investigated the blowup properties of the positive solutions. The aim of this paper is to study the uniqueness of the nonnegative solution of the problem (1) for any p
,
q
,
m
,
n
>
0.The plan of this paper is as follows. In the next section we prove a comparison principle; an existence theorem of a local solution is given in Section3;  uniqueness of solutions with nontrivial initial data, uniqueness of solution with trivial initial datum for min
⁡
(
p
,
q
,
m
,
n
)
≥
1; nonuniqueness of solution with trivial initial datum for min
⁡
(
p
q
,
m
,
n
)
<
1 are proved in Section 4.


## 2. Comparison Principle

Let us introduce the definitions of a subsolution and a supersolution. For the remainder of this paper we denoteQ
T
=
Ω
×
(
0
,
T
), S
T
=
∂
Ω
×
(
0
,
T
).Definition 1.
A pair of nonnegative functions(
u
,
v
)
∈
C
2,1
(
Q
T
)
⋂
C
(
Q
¯
T
) is called a subsolution of problem (1) in Q
T if
(2)
u
t
≤
Δ
u
+
c
1
(
x
,
t
)
v
p
,
v
t
≤
Δ
v
+
c
2
(
x
,
t
)
u
q
,
x
∈
Ω
,
0
<
t
<
T
,
u
(
x
,
t
)
≤
∫
Ω
k
1
(
x
,
y
,
t
)
u
m
(
y
,
t
)
d
y
,
x
∈
∂
Ω
,
0
<
t
<
T
,
v
(
x
,
t
)
≤
∫
Ω
k
2
(
x
,
y
,
t
)
v
n
(
y
,
t
)
d
y
,
x
∈
∂
Ω
,
0
<
t
<
T
,
u
(
x
,
0
)
≤
u
0
(
x
)
,
v
(
x
,
0
)
≤
v
0
(
x
)
,
x
∈
Ω
,
and a pair of nonnegative functions (
u
,
v
)
∈
C
2,1
(
Q
T
)
⋂
C
(
Q
¯
T
) is called a supersolution of problem (1) in Q
T if the reversed inequalities hold in (2).Definition 2.
A pair of functions(
u
,
v
) is called a solution of (1) in Q
T if it is both a subsolution and a supersolution of problem (1) in Q
T.Definition 3.
We say that solution(
u
,
v
) of (1) is positive in Q
T if u
>
0 and v
>
0 in Q
T.To establish the uniqueness results we need a comparison principle. We prove it in a different way, not as in the work of [7].Theorem 4.
Let(
u
¯
,
v
¯
) and (
u
_
,
v
_
) be a nonnegative supersolution and a nonnegative subsolution of problem (1) in Q
T, respectively. Suppose that u
¯
>
0 and v
¯
>
0 or u
_
>
0 and v
_
>
0 in Q
¯
T if min
⁡
(
p
,
q
,
m
,
n
)
<
1. If u
¯
(
x
,
0
)
≥
u
_
(
x
,
0
) and v
¯
(
x
,
0
)
≥
v
_
(
x
,
0
) for x
∈
Ω
¯, then u
¯
≥
u
_ and v
¯
≥
v
_ in Q
¯
T.Proof.
Letξ
(
x
,
t
)
∈
C
2,1
(
Q
¯
T
) be a nonnegative function such that ξ
|
S
T
=
0. Then ∂
ξ
/
∂
ν
|
S
T
≤
0, where ν is the unit outward normal to the lateral boundary of Q
T. By the definition of a subsolution we have
(3)
u
_
t
≤
Δ
u
_
+
c
1
(
x
,
t
)
v
_
p
,
x
∈
Ω
,
0
<
t
<
T
.
If we multiply (3) by ξ and then integrate over Q
t for 0
<
t
<
T, we get
(4)
∫
Ω
u
_
(
x
,
t
)
ξ
(
x
,
t
)
d
x
≤
∫
Ω
u
_
(
x
,
0
)
ξ
(
x
,
0
)
d
x
+
∫
0
t
∫
Ω
(
u
_
ξ
τ
+
u
_
Δ
ξ
+
c
1
(
x
,
τ
)
v
_
p
ξ
)
d
x
d
τ
-
∫
0
t
∫
∂
Ω
∂
ξ
∂
ν
(
∫
Ω
k
1
(
x
,
y
,
τ
)
u
_
m
(
y
,
τ
)
d
y
)
d
s
d
τ
.
On the other hand, the supersolutionu
¯ satisfies (4) with reversed inequality. Set w
(
x
,
t
)
=
u
_
(
x
,
t
)
-
u
¯
(
x
,
t
)
,
z
(
x
,
t
)
=
v
_
(
x
,
t
)
-
v
¯
(
x
,
t
). Then we have
(5)
∫
Ω
w
(
x
,
t
)
ξ
(
x
,
t
)
d
x
≤
∫
Ω
w
(
x
,
0
)
ξ
(
x
,
0
)
d
x
+
∫
0
t
∫
Ω
w
(
ξ
τ
+
Δ
ξ
)
d
x
d
τ
-
∫
0
t
∫
∂
Ω
∂
ξ
∂
ν
×
(
∫
Ω
k
1
(
x
,
y
,
τ
)
m
Θ
2
m
-
1
(
y
,
τ
)
w
(
y
,
τ
)
d
y
)
d
s
d
τ
+
∫
0
t
∫
Ω
z
(
x
,
t
)
p
c
1
(
x
,
t
)
Θ
1
p
-
1
ξ
d
x
d
τ
,
where Θ
1
,
Θ
2 are nonnegative continuous functions if min
⁡
(
p
,
q
,
m
,
n
)
≥
1 and positive continuous functions if min
⁡
(
p
,
q
,
m
,
n
)
<
1 in Q
¯
T which satisfy the following equalities:
(6)
v
_
p
-
v
¯
p
=
p
Θ
1
p
-
1
(
v
_
-
v
¯
)
,
u
_
m
-
u
¯
m
=
m
Θ
2
m
-
1
(
u
_
-
u
¯
)
.
Obviously there exists a positive constantM
1 such that Θ
1
p
-
1
≤
M
1
,
Θ
2
m
-
1
≤
M
1 in Q
¯
T. Since c
1
(
x
,
t
), k
1
(
x
,
y
,
t
) are nonnegative and continuous functions, then there exists a constant M
2
>
0 such that 0
≤
c
1
(
x
,
t
)
≤
M
2, 0
≤
k
1
(
x
,
y
,
t
)
≤
M
2 in Q
¯
T and ∂
Ω
×
Q
¯
T, respectively.
Consider the following backward problem inQ
t
:
(7)
ξ
τ
+
Δ
ξ
=
0
,
(
x
,
τ
)
∈
Q
t
,
ξ
=
0
,
(
x
,
τ
)
∈
S
t
,
ξ
(
x
,
t
)
=
ψ
(
x
)
,
x
∈
Ω
,
where ψ
(
x
)
∈
C
0
∞
(
Ω
)
,
0
≤
ψ
(
x
)
≤
1. By the maximum principle for the heat equation 0
≤
ξ
≤
1, it is easy to show that -
M
3
≤
∂
ξ
/
∂
ν
≤
0 on S
t for some M
3
≥
0.
Lets
+
=
max
⁡
(
s
,
0
). Then from (5), we get
(8)
∫
Ω
w
(
x
,
t
)
ψ
(
x
)
d
x
≤
K
1
∫
0
t
∫
Ω
z
+
(
x
,
t
)
d
x
d
t
≤
∫
Ω
w
(
x
,
t
)
ψ
(
x
)
d
x
+
K
2
∫
0
t
∫
Ω
w
+
(
y
,
τ
)
d
y
d
τ
,
where K
1
=
p
M
1
M
2
,
K
2
=
m
M
1
M
2
M
3
|
∂
Ω
|, and |
∂
Ω
| is the Lebesgue measure of ∂
Ω. Consider the sequence of functions {
ϕ
n
(
x
)
}, ϕ
n
(
x
)
∈
C
0
∞
(
Ω
¯
)
,
0
≤
ϕ
n
≤
1, converging in L
1
(
Ω
) to ϕ
(
x
), defined as follows:
(9)
ϕ
(
x
)
=
1
,
if
w
(
x
,
t
)
>
0
,
ϕ
(
x
)
=
0
,
if
w
(
x
,
t
)
≤
0
.
Replacing ψ
(
x
) by ϕ
n
(
x
) in (8) and passing to the limit as n
→
∞, we have
(10)
∫
Ω
w
+
(
x
,
t
)
d
x
≤
K
1
∫
0
t
∫
Ω
z
+
(
x
,
t
)
d
x
d
t
≤
∫
Ω
w
+
(
x
,
t
)
d
x
+
K
2
∫
0
t
∫
Ω
w
+
(
y
,
τ
)
d
y
d
τ
.
Using a similar argument for the inequality v
_
t
≤
Δ
v
_
+
c
2
(
x
,
t
)
u
_
q,  x
∈
Ω,  0
<
t
<
T, we get
(11)
∫
Ω
z
+
(
x
,
t
)
d
x
≤
K
3
∫
0
t
∫
Ω
w
+
(
x
,
t
)
d
x
d
t
∫
Ω
z
+
(
x
,
t
)
d
x
≤
+
K
4
∫
0
t
∫
Ω
z
+
(
y
,
τ
)
d
y
d
τ
,
where K
3, K
4 are positive constants. Adding (10) and (11), we have
(12)
∫
Ω
(
w
+
(
x
,
t
)
+
z
+
(
x
,
t
)
)
d
x
≤
K
5
∫
0
t
∫
Ω
(
w
+
(
x
,
t
)
+
z
+
(
x
,
t
)
)
d
x
d
t
,
where K
5
=
max
⁡
(
K
1
+
K
4
,
K
2
+
K
3
).
Applying now Gronwall's inequality, we conclude that(13)
∫
Ω
(
w
+
(
x
,
t
)
+
z
+
(
x
,
t
)
)
d
x
≤
0
.
Since w
+
+
z
+
≥
0, then u
_
(
x
,
t
)
-
u
¯
(
x
,
t
)
≤
0, v
_
(
x
,
t
)
-
v
¯
(
x
,
t
)
≤
0.


## 3. Local Existence

Let{
ε
l
} be decreasing to 0 sequence such that 0
<
ε
l
<
1. For ɛ
=
ε
l let u
0
ɛ
(
x
)
,
v
0
ɛ
(
x
) be the functions with the following properties: u
0
ɛ
(
x
)
,
v
0
ɛ
(
x
)
∈
C
(
Ω
¯
)
,
u
0
ɛ
(
x
)
≥
ɛ
,
v
0
ɛ
(
x
)
≥
ɛ; u
0
ε
i
(
x
)
≥
u
0
ε
j
(
x
)
,
v
0
ε
i
(
x
)
≥
v
0
ε
j
(
x
) for i
<
j; u
0
ɛ
(
x
)
→
u
0
(
x
)
,
v
0
ɛ
(
x
)
→
v
0
(
x
) as ɛ
→
0; u
0
ɛ
(
x
)
=
∫
Ω
‍
k
1
(
x
,
y
,
0
)
u
0
ɛ
m
(
y
)
d
y
+
ɛ
,
v
0
ɛ
(
x
)
=
∫
Ω
‍
k
2
(
x
,
y
,
0
)
v
0
ɛ
n
(
y
)
d
y
+
ɛ for x
∈
∂
Ω.Due to the nonlinearities in (1), the Lipschitz condition is not satisfied if min
⁡
⁡
(
p
,
q
,
m
,
n
)
<
1, and thus we need to consider the following auxiliary problem:
(14)
u
t
=
Δ
u
+
c
1
(
x
,
t
)
v
p
,
v
t
=
Δ
v
+
c
2
(
x
,
t
)
u
q
,
x
∈
Ω
,
0
<
t
<
T
,
u
(
x
,
t
)
=
∫
Ω
k
1
(
x
,
y
,
t
)
u
m
(
y
,
t
)
d
y
+
ε
,
x
∈
∂
Ω
,
0
<
t
<
T
,
v
(
x
,
t
)
=
∫
Ω
k
2
(
x
,
y
,
t
)
v
n
(
y
,
t
)
d
y
+
ε
,
x
∈
∂
Ω
,
0
<
t
<
T
,
u
(
x
,
0
)
=
u
0
ε
(
x
)
,
v
(
x
,
0
)
=
v
0
ε
(
x
)
,
x
∈
Ω
.Theorem 5.
For small values ofT, (14) has a unique solution in Q
T.Proof.
We start the proof with the construction of a supersolution of (14). Let max
⁡
(
su
p
Ω
u
0
ɛ
(
x
)
,
su
p
Ω
v
0
ɛ
(
x
)
)
≤
C,  C
>
0. Denote M
=
max
⁡
(
su
p
Q
T
c
1
(
x
,
t
)
,
su
p
Q
T
c
2
(
x
,
t
)
,
su
p
∂
Ω
×
Q
T
k
1
(
x
,
y
,
t
)
,
su
p
∂
Ω
×
Q
T
k
2
(
x
,
y
,
t
)
). Introduce an auxiliary function   φ
(
x
) with the following properties:
(15)
φ
(
x
)
∈
C
2
(
Ω
¯
)
,
inf
⁡
Ω
⁡
φ
(
x
)
≥
1
,
inf
⁡
∂
Ω
⁡
φ
(
x
)
≥
M
¯
∫
Ω
max
⁡
⁡
(
φ
m
(
y
)
,
φ
n
(
y
)
)
d
y
+
C
-
1
,
where  M
¯
=
M
max
⁡
(
C
m
-
1
,
C
n
-
1
)
max
⁡
(
1
,
exp
⁡
(
m
-
1
)
,
exp
⁡
(
n
-
1
)
). Let α
,
β be positive constants such that α
q
-
β
=
β
p
-
α and
(16)
α
≥
sup
⁡
Ω
⁡
(
Δ
φ
φ
+
M
C
p
-
1
exp
⁡
(
1
)
φ
p
-
1
)
,
β
≥
sup
⁡
Ω
⁡
(
Δ
φ
φ
+
M
C
q
-
1
exp
⁡
(
1
)
φ
q
-
1
)
.
Note that α
q
-
β
>
0 for p
q
>
1 and α
q
-
β
≤
0 for p
q
≤
1. Obviously the pair of functions (
ε
,
ε
) is a subsolution of problem (14). We show that
(17)
f
(
x
,
t
)
=
C
exp
⁡
⁡
(
α
t
)
φ
(
x
)
,
g
(
x
,
t
)
=
C
exp
⁡
⁡
(
β
t
)
φ
(
x
)
is a supersolution of (14) in Q
T for T
≤
min
⁡
(
1
/
α
,
1
/
β
,
1
/
(
α
q
-
β
)
) if p
q
>
1 and T
≤
min
⁡
(
1
/
α
,
1
/
β
) if p
q
≤
1. We have
(18)
f
t
(
x
,
t
)
-
Δ
f
(
x
,
t
)
-
c
1
(
x
,
t
)
g
p
(
x
,
t
)
=
α
C
exp
⁡
(
α
t
)
φ
(
x
)
-
Δ
φ
C
exp
⁡
(
α
t
)
-
c
1
(
x
,
t
)
C
p
exp
⁡
⁡
(
β
p
t
)
φ
p
(
x
)
≥
f
(
x
,
t
)
(
α
-
Δ
φ
(
x
)
φ
(
x
)
-
M
C
p
-
1
exp
⁡
⁡
(
(
β
p
-
α
)
t
)
φ
p
-
1
)
≥
0
,
for (
x
,
t
)
∈
Q
T. On the other hand, we get that
(19)
C
exp
⁡
(
α
t
)
φ
(
x
)
≥
C
m
exp
⁡
(
α
m
t
)
M
∫
Ω
φ
m
(
y
)
d
y
+
1
C
exp
⁡
(
α
t
)
φ
(
x
)
≥
∫
Ω
k
1
(
x
,
y
,
t
)
f
m
(
y
,
t
)
d
y
+
ε
,
for (
x
,
t
)
∈
S
T. Similarly, we can show that
(20)
g
t
(
x
,
t
)
-
Δ
g
(
x
,
t
)
-
c
2
(
x
,
t
)
f
q
(
x
,
t
)
≥
0
,
for
(
x
,
t
)
∈
Q
T
,
g
(
x
,
t
)
≥
∫
Ω
k
2
(
x
,
y
,
t
)
g
n
(
y
,
t
)
d
y
+
ε
,
for
(
x
,
t
)
∈
S
T
.
To prove the existence of a solution of the problem (14) we introduce the set
(21)
B
=
{
(
h
1
,
h
2
)
∈
C
(
Q
¯
T
)
×
C
(
Q
¯
T
)
:
ε
≤
h
1
≤
f
,
ε
≤
h
2
≤
g
,
h
1
(
x
,
0
)
=
u
0
ε
(
x
)
,
h
2
(
x
,
0
)
=
v
0
ε
(
x
)
(
Q
¯
T
)
}
.
Clearly B is a nonempty convex subset of C
(
Q
¯
T
)
×
C
(
Q
¯
T
).
Consider the following problem:(22)
u
t
=
Δ
u
+
c
1
(
x
,
t
)
v
p
,
v
t
=
Δ
v
+
c
2
(
x
,
t
)
u
q
,
x
∈
Ω
,
0
<
t
<
T
,
u
(
x
,
t
)
=
∫
Ω
k
1
(
x
,
y
,
t
)
s
1
m
(
y
,
t
)
d
y
+
ε
,
x
∈
∂
Ω
,
0
<
t
<
T
,
v
(
x
,
t
)
=
∫
Ω
k
2
(
x
,
y
,
t
)
s
2
n
(
y
,
t
)
d
y
+
ε
,
x
∈
∂
Ω
,
0
<
t
<
T
,
u
(
x
,
0
)
=
u
0
ɛ
(
x
)
,
v
(
x
,
0
)
=
v
0
ɛ
(
x
)
,
x
∈
Ω
,
where (
s
1
,
s
2
)
∈
B. Problem (22) has a nontrivial positive solution. Let us call A
(
s
1
,
s
2
)
=
(
u
,
v
). In order to show that A has a fixed point in B we verify that A is a continuous mapping from B into itself such that A
B is relatively compact. Thanks to the comparison principle for (22) we have that A maps B into itself.
LetG
(
x
,
y
;
t
) denote the Green's function for the heat equation given by
(23)
u
t
-
Δ
u
=
0
,
x
∈
Ω
,
t
>
0
with zero boundary condition. Then (
u
,
v
) is a solution of (22) if and only if
(24)
u
(
x
,
t
)
=
∫
Ω
G
(
x
,
y
;
t
)
u
0
ε
(
y
)
d
y
+
∫
0
t
∫
Ω
G
(
x
,
y
;
t
-
τ
)
c
1
(
y
,
τ
)
v
p
(
y
,
τ
)
d
y
d
τ
-
∫
0
t
∫
∂
Ω
∂
G
∂
ν
(
x
,
ξ
;
t
-
τ
)
×
(
∫
Ω
k
1
(
ξ
,
y
,
τ
)
s
1
m
(
y
,
τ
)
d
y
+
ε
)
d
ξ
d
τ
,
(25)
v
(
x
,
t
)
=
∫
Ω
G
(
x
,
y
;
t
)
v
0
ɛ
(
y
)
d
y
+
∫
0
t
∫
Ω
G
(
x
,
y
;
t
-
τ
)
c
2
(
y
,
τ
)
u
q
(
y
,
τ
)
d
y
d
τ
-
∫
0
t
∫
∂
Ω
∂
G
∂
ν
(
x
,
ξ
;
t
-
τ
)
×
(
∫
Ω
k
2
(
ξ
,
y
,
τ
)
s
2
n
(
y
,
τ
)
d
y
+
ε
)
d
ξ
d
τ
.
We claim thatA is continuous. In fact, let {
(
s
1
k
,
s
2
k
)
} be a sequence in B converging to (
s
1
,
s
2
)
∈
B in C
(
Q
¯
T
)
×
C
(
Q
¯
T
). Denote (
u
k
,
v
k
)
=
A
(
s
1
k
,
s
2
k
). Then we see that
(26)
|
u
-
u
k
|
+
|
v
-
v
k
|
≤
∫
0
t
∫
Ω
G
(
x
,
y
;
t
-
τ
)
c
1
(
y
,
τ
)
|
v
p
-
v
k
p
|
d
y
d
τ
‍
‍
+
∫
0
t
∫
Ω
G
(
x
,
y
;
t
-
τ
)
c
2
(
y
,
τ
)
|
u
q
-
u
k
q
|
d
y
d
τ
‍
‍
-
sup
⁡
Q
T
|
s
1
m
-
s
1
k
m
|
∫
0
t
∫
∂
Ω
∂
G
∂
ν
(
x
,
ξ
;
t
-
τ
)
‍
‍
×
∫
Ω
k
1
(
ξ
,
y
,
τ
)
d
y
d
ξ
d
τ
‍
-
sup
⁡
Q
T
|
s
2
n
-
s
2
k
n
|
∫
0
t
∫
∂
Ω
∂
G
∂
ν
(
x
,
ξ
;
t
-
τ
)
‍
‍
×
∫
Ω
k
2
(
ξ
,
y
,
τ
)
‍
d
y
d
ξ
d
τ
≤
r
(
sup
⁡
Q
T
⁡
|
u
-
u
k
|
+
sup
⁡
Q
T
⁡
|
v
-
v
k
|
)
-
sup
⁡
Q
T
|
s
1
m
-
s
1
k
m
|
∫
0
t
∫
∂
Ω
∂
G
∂
ν
(
x
,
ξ
;
t
-
τ
)
‍
‍
×
∫
Ω
k
1
(
ξ
,
y
,
τ
)
d
y
d
ξ
d
τ
-
sup
⁡
Q
T
|
s
2
n
-
s
2
k
n
|
∫
0
t
∫
∂
Ω
∂
G
∂
ν
‍
(
x
,
ξ
;
t
-
τ
)
×
‍
∫
Ω
k
2
(
ξ
,
y
,
τ
)
d
y
d
ξ
d
τ
,
where r
=
max
⁡
(
Θ
1
,
Θ
2
) and
(27)
Θ
1
=
p
max
⁡
⁡
(
ε
p
-
1
,
sup
⁡
Q
T
h
1
p
-
1
(
x
,
t
)
)
×
sup
⁡
Q
T
∫
0
t
∫
Ω
G
(
x
,
y
;
t
-
τ
)
c
1
(
y
,
τ
)
d
y
d
τ
,
‍
Θ
2
=
q
max
⁡
⁡
(
ε
q
-
1
,
sup
⁡
Q
T
h
2
q
-
1
(
x
,
t
)
)
×
sup
⁡
Q
T
∫
0
t
∫
Ω
G
(
x
,
y
;
t
-
τ
)
c
2
(
y
,
τ
)
d
y
d
τ
.
‍
‍
Choosing T so small that r
<
1, we conclude that (
u
k
,
v
k
)
→
(
u
,
v
) in C
(
Q
¯
T
)
×
C
(
Q
¯
T
) as k
→
∞. The equicontinuity of A
B follows from (24), (25) and the properties of the Green's function (see, e.g. [22]). The Ascoli-Arzela theorem guarantees the relative compactness of A
B. Thus, we are able to apply the Schauder-Tychonoff fixed point theorem and conclude that A has a fixed point in B if T is small. Since (
u
,
v
) is a fixed point of A, it is a solution of (14). Uniqueness of solution follows from a comparison principle for (14) which can be proved in a similar way as in the previous section.Using Theorem5, we can prove the following local existence theorem of a solution of problem (1).Theorem 6.
For small values ofT (1) has a maximal solution in Q
T.Proof.
Letε
2
>
ε
1. It is easy to show that (
u
ε
2
(
x
,
t
)
,
v
ε
2
(
x
,
t
)
) is a supersolution of the problem (14) with ε
=
ε
1. Then u
ε
1
(
x
,
t
)
≤
u
ε
2
(
x
,
t
),  v
ε
1
(
x
,
t
)
≤
v
ε
2
(
x
,
t
). Using these inequalities and the continuation principle of solutions we deduce that the existence time of (
u
ɛ
(
x
,
t
)
,
v
ɛ
(
x
,
t
)
) does not decrease as ε
→
0. Let ε
→
0, then
(28)
u
max
⁡
(
x
,
t
)
=
lim
⁡
ɛ
→
0
⁡
u
ε
(
x
,
t
)
≥
0
,
v
max
⁡
(
x
,
t
)
=
lim
⁡
ε
→
0
⁡
v
ɛ
(
x
,
t
)
≥
0
,
and (
u
max
⁡
(
x
,
t
)
,
v
max
⁡
(
x
,
t
)
) exist in Q
T for some T
>
0.
Moreover, by dominated convergence theorem,(
u
max
⁡
(
x
,
t
)
,
v
max
⁡
(
x
,
t
)
) satisfies the following equations:
(29)
u
max
⁡
(
x
,
t
)
=
∫
Ω
G
(
x
,
y
;
t
)
u
0
(
y
)
d
y
‍
+
∫
0
t
∫
Ω
G
(
x
,
y
;
t
-
τ
)
c
1
(
y
,
τ
)
‍
×
v
max
⁡
p
(
y
,
τ
)
d
y
d
τ
-
∫
0
t
∫
∂
Ω
∂
G
∂
ν
(
x
,
ξ
;
t
-
τ
)
‍
‍
×
(
∫
Ω
k
1
(
ξ
,
y
,
τ
)
u
max
⁡
m
(
y
,
τ
)
d
y
‍
)
d
ξ
d
τ
,
v
max
⁡
(
x
,
t
)
=
∫
Ω
G
(
x
,
y
;
t
)
v
0
(
y
)
d
y
‍
+
∫
0
t
∫
Ω
G
(
x
,
y
;
t
-
τ
)
‍
c
2
(
y
,
τ
)
×
u
max
⁡
q
(
y
,
τ
)
d
y
d
τ
-
∫
0
t
∫
∂
Ω
∂
G
∂
ν
(
x
,
ξ
;
t
-
τ
)
‍
‍
×
(
∫
Ω
k
2
(
ξ
,
y
,
τ
)
v
max
⁡
n
(
y
,
τ
)
d
y
‍
)
d
ξ
d
τ
.
The interior regularity of (
u
max
⁡
(
x
,
t
)
,
v
max
⁡
(
x
,
t
)
) follows from the continuity of (
u
max
⁡
(
x
,
t
)
,
v
max
⁡
(
x
,
t
)
) in Q
T and the properties of the Green's function. Obviously (
u
max
⁡
(
x
,
t
)
,
v
max
⁡
(
x
,
t
)
) satisfies (1). Let (
y
1
(
x
,
t
)
,
y
2
(
x
,
t
)
) be any other solution of (1). Then by the comparison principle u
ε
(
x
,
t
)
≥
y
1
(
x
,
t
)
,
v
ε
(
x
,
t
)
≥
y
2
(
x
,
t
). Taking ε
→
0, we conclude that u
max
⁡
≥
y
1
(
x
,
t
)
,
v
max
⁡
(
x
,
t
)
≥
y
2
(
x
,
t
).To prove the positiveness of nontrivial solutions we need the following definition.Definition 7.
We say that a functiong
(
x
,
t
) has the property (N) if there exist x
k
∈
Ω and t
k
>
0,k
∈
ℕ such that g
(
x
k
,
t
k
)
>
0,  k
∈
ℕ, and t
k
→
0 as k
→
∞.Remark 8.
Note that if a nonnegative functiong
(
x
,
t
) has no the property (N) then g
(
x
,
t
)
≡
0 in Q
τ for some τ
>
0.Theorem 9.
Let eitheru
0
(
x
) or v
0
(
x
) be a nontrivial function in Ω. Supposing that k
1
(
x
,
·
,
t
) and k
2
(
x
,
·
,
t
) are nontrivial functions for any x
∈
∂
Ω and t
∈
(
0
,
T
),  c
1
(
x
,
t
) has the property (N) if u
0
(
x
)
≡
0,  and  c
2
(
x
,
t
) has the property (N) if v
0
(
x
)
≡
0. Let (
u
(
x
,
t
)
,
v
(
x
,
t
)
) be a supersolution of (1) in Q
T. Then (
u
(
x
,
t
)
,
v
(
x
,
t
)
) is positive in Q
¯
T for 0
<
t
<
T.Proof.
Suppose for definiteness thatu
0
(
x
) is a nontrivial function. We show at first that u
(
x
,
t
)
>
0 in Q
¯
T for 0
<
t
<
T. We have
(30)
u
t
-
Δ
u
≥
c
1
(
x
,
t
)
v
p
≥
0
;
then by strong maximum principle a minimum of u
(
x
,
t
) should be attained in Q
¯
T on a parabolic boundary. Thus, u
(
x
,
t
)
>
0 in Q
T, otherwise, there would be a contradiction with the initial datum. We show that u
(
x
,
t
)
>
0 on ∂
Ω
×
(
0
,
T
). Let there exist a point (
x
0
,
t
0
)
∈
S
T such that u
(
x
0
,
t
0
)
=
0. But u
(
x
,
t
0
)
>
0 for x
∈
Ω. By boundary conditions (1) and assumption for k
1
(
x
,
y
,
t
) we have u
(
x
,
t
)
>
0 for (
x
,
t
)
∈
S
T. This contradiction shows that u
(
x
,
t
)
>
0 on S
T, and therefore u
(
x
,
t
)
>
0 in Q
¯
T for 0
<
t
<
T.
Now we show the positiveness ofv
(
x
,
t
). If v
0
(
x
) is a nontrivial function, then v
(
x
,
t
)
>
0 in Q
¯
T for 0
<
t
<
T by previous arguments. If v
0
(
x
)
≡
0 we suppose that there exists a constant τ
>
0 such that v
(
x
,
t
)
≡
0 in Q
τ since otherwise we can use the arguments from the beginning of the proof again. But this is a contradiction with the second equation in (1) since u
(
x
,
t
)
>
0 in Q
τ and c
2
(
x
,
t
) has the property (N). Hence, we conclude that v
(
x
,
t
)
>
0 in Q
¯
T for 0
<
t
<
T.


## 4. Uniqueness and Nonuniqueness

As a simple consequence of Theorem4 and Theorem 9 we get the first uniqueness result for problem (1).Theorem 10.
Let problem (1) have a solution in Q
T with nonnegative initial datum for min
⁡
(
p
,
q
,
m
,
n
)
≥
1 and with positive initial datum under conditions min
⁡
(
p
,
q
,
m
,
n
)
<
1 and k
1
(
x
,
·
,
t
)
,
k
2
(
x
,
·
,
t
) are nontrivial functions for any x
∈
∂
Ω and t
∈
(
0
,
T
). Then solution of (1) is unique in Q
T.Now we show nonuniqueness of solutions of problem (1) with trivial initial datum for min
⁡
(
p
q
,
m
,
n
)
<
1.Theorem 11.
Letmin
⁡
(
p
q
,
m
,
n
)
<
1, u
0
(
x
)
=
v
0
(
x
)
≡
0. Suppose that the maximal solution of problem (1) exists in Q
T. Assume that at least one of the following conditions is fulfilled:
(31)
c
1
(
x
0
,
t
0
)
>
0
,
c
2
(
x
0
,
t
0
)
>
0
f
o
r
s
o
m
e
x
0
∈
Ω
,
t
0
∈
[
0
,
T
)
i
f
p
q
<
1
,
(32)
k
1
(
x
,
y
1
,
t
1
)
>
0
f
o
r
a
n
y
x

∈

∂
Ω
,
a
n
d
s
o
m
e
y
1

∈

∂
Ω
,
t
1

∈

[
0
,
T
)
i
f
m
<
1
,
(33)
k
2
(
x
,
y
2
,
t
2
)
>
0
f
o
r
a
n
y
x

∈

∂
Ω
,
a
n
d
s
o
m
e
y
2

∈

∂
Ω
,
t
2

∈

[
0
,
T
)
i
f
n
<
1
.
Then the maximal solution of problem (1) is nontrivial in Q
T.Proof.
In the local existence theorem we constructed a maximal solution(
u
max
⁡
(
x
,
t
)
,
v
max
⁡
(
x
,
t
)
) of (1) in the following way: u
max
⁡
(
x
,
t
)
=
li
m
ε
→
0
u
ε
(
x
,
t
), v
max
⁡
(
x
,
t
)
=
li
m
ε
→
0
v
ε
(
x
,
t
), where (
u
ε
(
x
,
t
)
,
v
ε
(
x
,
t
)
) is some positive supersolution of (1). To prove the theorem we construct a nontrivial nonnegative subsolution (
u
_
(
x
,
t
)
,
v
_
(
x
,
t
)
) of some problem with trivial initial datum. By the comparison principle we conclude that u
ε
(
x
,
t
)
≥
u
_
(
x
,
t
)
,
v
ε
(
x
,
t
)
)
≥
v
_
(
x
,
t
), and therefore maximal solution is a nontrivial solution.
Consider at first the case whenp
q
<
1 and c
1
(
x
0
,
t
0
)
>
0
,
c
2
(
x
0
,
t
0
)
>
0 for some x
0
∈
Ω and 0
≤
t
0
<
T. Then there exists a neighborhood U
(
x
0
) of x
0 in Ω and t
¯
∈
(
t
0
,
T
) such that c
1
(
x
,
t
)
≥
c
0
,
c
2
(
x
,
t
)
≥
c
0
,
c
0
>
0 for x
∈
U
(
x
0
)
and
t
0
≤
t
≤
t
¯.
Consider the following problem:(34)
w
t
=
Δ
w
,
x
∈
U
(
x
0
)
,
t
0
<
t
<
t
¯
,
w
(
x
,
t
)
=
0
,
x
∈
∂
U
(
x
0
)
,
t
0
<
t
<
t
¯
,
w
(
x
,
t
0
)
=
w
0
(
x
)
,
x
∈
U
(
x
0
)
,
where w
0
(
x
) is a bounded nontrivial nonnegative continuous function which satisfies a boundary condition. By the strong maximum principle 0
<
w
(
x
,
t
)
<
M
0 for t
∈
(
t
0
,
t
-
)
,
x
∈
U
(
x
0
), where M
0
=
sup
⁡
x
∈
U
(
x
0
)
w
0
(
x
).
Letα
=
(
1
+
p
)
/
(
1
-
p
q
)
>
1
,
β
=
(
1
+
q
)
(
1
-
p
q
)
>
1. Denote that
(35)
u
_
(
x
,
t
)
=
C
α
(
t
-
t
0
)
α
w
α
(
x
,
t
)
,
v
_
(
x
,
t
)
=
C
β
(
t
-
t
0
)
β
w
β
(
x
,
t
)
,
where C
=
min
⁡
(
c
0
/
(
α
M
0
)
,
c
0
/
(
β
M
0
)
). Note that u
_
(
x
,
t
)
=
0
,
v
_
(
x
,
t
)
=
0 if t
=
t
0 or x
∈
∂
U
(
x
0
). After simple calculations we obtain
(36)
u
_
t
-
Δ
u
_
-
c
1
(
x
,
t
)
v
_
p
=
α
C
α
(
t
-
t
0
)
α
-
1
w
α
+
α
C
α
(
t
-
t
0
)
α
w
α
-
1
w
t
-
α
Δ
w
w
α
-
1
C
α
(
t
-
t
0
)
α
-
α
(
α
-
1
)
w
α
-
2
|
∇
w
|
2
C
α
(
t
-
t
0
)
α
-
c
1
(
x
,
t
)
C
β
p
(
t
-
t
0
)
β
p
w
β
p
≤
C
β
p
(
t
-
t
0
)
β
p
w
β
p
(
α
C
w
-
c
1
(
x
,
t
)
)
≤
0
.
Similarly we can get that
(37)
v
_
t
-
Δ
v
_
-
c
2
(
x
,
t
)
u
_
q
≤
0
.
Then (
u
_
(
x
,
t
)
,
v
_
(
x
,
t
)
) and (
u
ε
(
x
,
t
)
,
v
ε
(
x
,
t
)
) are subsolution and supersolution, respectively, of the following problem:
(38)
u
t
=
Δ
u
+
c
1
(
x
,
t
)
v
p
,
v
t
=
Δ
v
+
c
2
(
x
,
t
)
u
q
,
x
∈
U
(
x
0
)
,
t
0
<
t
<
t
¯
,
u
(
x
,
t
)
=
v
(
x
,
t
)
=
0
,
x
∈
∂
U
(
x
0
)
,
t
0
<
t
<
t
¯
,
u
(
x
,
t
0
)
=
v
(
x
,
t
0
)
=
0
,
x
∈
U
(
x
0
)
.
By comparison principle for (38) we conclude that u
ε
(
x
,
t
)
≥
u
_
(
x
,
t
), v
ε
(
x
,
t
)
≥
v
_
(
x
,
t
) and hence u
max
⁡
(
x
,
t
)
≥
u
_
(
x
,
t
),  v
max
⁡
(
x
,
t
)
≥
v
_
(
x
,
t
), for (
x
,
t
)
∈
U
(
x
0
)
×
(
t
0
,
t
¯
).
Now consider the case whenm
<
1 and k
1
(
x
,
y
1
,
t
1
)
>
0 for any x
∈
∂
Ω and some y
1
∈
∂
Ω,  0
≤
t
1
<
T. We will consider the following problem:
(39)
u
t
=
Δ
u
+
c
1
(
x
,
t
)
v
p
,
v
t
=
Δ
v
+
c
2
(
x
,
t
)
u
q
,
x
∈
Ω
,
t
1
<
t
<
T
0
,
u
(
x
,
t
)
=
∫
Ω
k
1
(
x
,
y
,
t
)
u
m
(
y
,
t
)
d
y
‍
,
x
∈
∂
Ω
,
t
1
<
t
<
T
0
,
v
(
x
,
t
)
=
0
,
x
∈
∂
Ω
,
t
1
<
t
<
T
0
,
u
(
x
,
t
1
)
=
0
,
v
(
x
,
t
1
)
=
0
,
x
∈
Ω
,
where T
0
>
t
1 will be defined later. Construct a subsolution of (39) using the change of variables in a neighborhood of ∂
Ω as in [23]. Let x
¯
∈
∂
Ω. We denote by n
^ the inner unit normal to ∂
Ω at the point x
¯. Since ∂
Ω is smooth it is well known that there exists δ
>
0 such that the mapping ψ
:
∂
Ω
×
[
0
,
δ
]
→
ℝ
N given by ψ
(
x
¯
,
s
)
=
x
¯
+
s
n
^
(
x
¯
) defines new coordinates (
x
¯
,
s
) in a neighborhood of ∂
Ω in Ω
¯. A straightforward computation shows that, in these coordinates, Δ applied to a function g
(
x
¯
,
s
)
=
g
(
s
) which is independent of the variable x
¯, evaluated at a point (
x
¯
,
s
) is given by
(40)
Δ
g
(
x
¯
,
s
)
=
∂
2
g
∂
s
2
(
x
¯
,
s
)
-
∑
j
=
1
N
-
1
H
j
(
x
¯
)
1
-
s
H
j
(
x
¯
)
∂
g
∂
s
‍
(
x
¯
,
s
)
,
where H
j
(
x
¯
)
,
j
=
1
,
…
,
N
-
1 denote the principal curvatures of ∂
Ω at x
¯.
Under the made assumption there existst
~
>
0 such that k
1
(
x
,
y
,
t
)
>
0 for t
1
≤
t
≤
t
1
+
t
~
,
x
∈
∂
Ω
,
y
∈
V
(
y
1
), where V
(
y
1
) is some neighborhood of y
1 in Ω
¯. Let α
>
1
/
(
2
(
1
-
m
)
) and assume that 0
<
ξ
0
≤
1 and t
1
<
T
0
≤
t
1
+
min
⁡
(
t
~
,
δ
2
,
T
-
t
1
). For points in ∂
Ω
×
[
0
,
δ
]
×
(
t
1
,
T
0
] of coordinates (
x
¯
,
s
,
t
) define
(41)
u
_
(
x
¯
,
s
,
t
)
=
(
t
-
t
1
)
α
(
ξ
0
-
s
t
-
t
1
)
+
3
,
v
_
(
x
¯
,
s
,
t
)
=
0
and extend u
_
,
v
_ as zero to the whole Ω
¯
×
[
t
1
,
T
0
]. Using (40), we get that
(42)
u
_
t
(
x
¯
,
s
,
t
)
-
Δ
u
_
(
x
¯
,
s
,
t
)
-
c
1
(
x
,
t
)
v
_
p
(
x
¯
,
s
,
t
)
=
3
s
(
t
-
t
1
)
α
-
3
/
2
2
(
ξ
0
-
s
t
-
t
1
)
+
2
+
α
(
t
-
t
1
)
α
-
1
(
ξ
0
-
s
t
-
t
1
)
+
3
-
6
(
t
-
t
1
)
α
-
1
(
ξ
0
-
s
t
-
t
1
)
+
-
3
(
t
-
t
1
)
α
-
1
/
2
(
ξ
0
-
s
t
-
t
1
)
+
2
×
∑
j
=
1
N
-
1
H
j
(
x
¯
)
1
-
s
H
j
(
x
¯
)
‍
≤
0
for sufficiently small values of ξ
0. It is clear that
(43)
v
_
t
(
x
¯
,
s
,
t
)
-
Δ
v
_
(
x
¯
,
s
,
t
)
-
c
2
(
x
,
t
)
u
_
q
(
x
¯
,
s
,
t
)
≤
0
.
It remains to verify the validity of the inequality
(44)
(
t
-
t
1
)
α
ξ
0
3
≤
(
t
-
t
1
)
α
m
‍
×
∫
∂
Ω
×
[
0
,
δ
]
k
1
(
x
,
(
y
¯
,
s
)
,
t
)
|
J
(
y
¯
,
s
)
|
×
(
ξ
0
-
s
t
-
t
1
)
+
3
m
d
y
¯
d
s
for x
∈
∂
Ω and t
1
<
t
<
T
0. Here J
(
y
¯
,
s
) is Jacobian of the change of variables. Estimating the integral I in the right-hand side of (44) we get
(45)
I
=
(
t
-
t
1
)
1
/
2
×
∫
∂
Ω
d
y
¯
∫
0
ξ
0
k
1
(
x
,
(
y
¯
,
z
t
-
t
1
)
,
t
)
×
|
J
(
y
¯
,
z
t
-
t
1
)
|
(
ξ
0
-
z
)
+
3
m
d
z
≥
C
(
t
-
t
1
)
1
/
2
,
where constant C does not depend on t; we obtain that (44) is true if T
0
-
t
1 is small enough.
By comparison principle for (39) we conclude that u
ɛ
(
x
,
t
)
≥
u
_
(
x
,
t
), v
ε
(
x
,
t
)
≥
v
_
(
x
,
t
) and, respectively, u
max
⁡
(
x
,
t
)
≥
u
_
(
x
,
t
),  v
max
⁡
(
x
,
t
)
≥
v
_
(
x
,
t
) for (
x
,
t
)
∈
Ω
×
(
t
1
,
T
0
).
The proof in the casen
<
1 and k
2
(
x
,
y
2
,
t
2
)
>
0 for any x
∈
∂
Ω and some y
2
∈
∂
Ω
,
0
≤
t
2
<
T is similar.It is easy to get from Theorem9 and the proof of Theorem 11 the following statement.Corollary 12.
Let the conditions of Theorem11 hold with t
i
=
0
,
i
=
0,1
,
2. Suppose also that k
1
(
x
,
·
,
t
) and k
2
(
x
,
·
,
t
) are nontrivial functions for any x
∈
∂
Ω and t
∈
(
0
,
T
),   c
2
(
x
,
t
) has the property (N) if (32) is realized, and c
1
(
x
,
t
) has the property (N) if (33) is realized. Then maximal solution is positive in Q
¯
T for 0
<
t
<
T.Under the conditions of Corollary12 for some class of the coefficients c
i
(
x
,
t
) and k
i
(
x
,
y
,
t
),   i
=
1,2, we can prove the uniqueness of solution for (1) with trivial initial datum which is positive for all positive times as long as it exists.Theorem 13.
Let the conditions of Corollary12 hold. Suppose also that there exists t
0
>
0 such that for 0
≤
t
≤
t
0 the functions c
1
(
x
,
t
), c
2
(
x
,
t
),  k
1
(
x
,
y
,
t
)
,
a
n
d
k
2
(
x
,
y
,
t
) are nondecreasing with respect to t.
Then there exists exactly one solution of problem (1) which is positive in Q
¯
T for 0
<
t
<
T.Proof.
Suppose that there exists different from(
u
max
⁡
(
x
,
t
)
,
v
max
⁡
(
x
,
t
)
) solution (
u
(
x
,
t
)
,
v
(
x
,
t
)
) of (1) with trivial initial datum which is a positive in Q
¯
T for 0
<
t
<
T. Denote t
*
=
min
⁡
(
t
0
,
T
). Due to the conditions of the theorem it is easy to see that u
(
x
,
t
+
τ
)
,
v
(
x
,
t
+
τ
) is positive supersolution of (1) with trivial initial datum in Q
t
*
-
τ for any τ
∈
(
0
,
t
*
). By Theorem 4 we have u
max
⁡
(
x
,
t
)
≤
u
(
x
,
t
+
τ
)
,
v
max
⁡
(
x
,
t
)
≤
v
(
x
,
t
+
τ
) for every 0
≤
t
≤
t
*
-
τ. Passing to the limit as τ
→
0 we get u
max
⁡
(
x
,
t
)
≤
u
(
x
,
t
)
,
v
max
⁡
(
x
,
t
)
≤
v
(
x
,
t
) for 0
≤
t
≤
t
*. Hence u
max
⁡
(
x
,
t
)
=
u
(
x
,
t
)
,
v
max
⁡
(
x
,
t
)
=
v
(
x
,
t
) in Q
T.Note that by Theorem10, the solution (
u
(
x
,
t
)
,
v
(
x
,
t
)
) of (1) is unique if min
⁡
(
p
,
q
,
m
,
n
)
≥
1. Now we specify our uniqueness result in the case min
⁡
(
p
q
,
m
,
n
)
<
1.Theorem 14.
Let the conditions of Corollary12 fulfill only u
0
(
x
)
≢
0 or v
0
(
x
)
≢
0. Then the solution of (1) is unique.Proof.
To prove the uniqueness of the solution ifmin
⁡
(
p
q
,
m
,
n
)
<
1, it suffices to show that if (
u
(
x
,
t
)
,
v
(
x
,
t
)
) is any solution of (1), then
(46)
u
max
⁡
(
x
,
t
)
≤
u
(
x
,
t
)
,
v
max
⁡
(
x
,
t
)
≤
v
(
x
,
t
)
,
where (
u
max
⁡
(
x
,
t
)
,
v
max
⁡
(
x
,
t
)
) is a maximal solution of (1).
First, consider the case when0
<
p
<
1
,
0
<
q
≤
1
,
0
<
m
<
1
,
and
0
<
n
≤
1. Let
(47)
w
1
(
x
,
t
)
=
u
max
⁡
(
x
,
t
)
-
u
(
x
,
t
)
,
w
2
(
x
,
t
)
=
v
max
⁡
(
x
,
t
)
-
v
(
x
,
t
)
.
Then (
w
1
,
w
2
) satisfies
(48)
w
1
t
≤
Δ
w
1
+
c
1
(
x
,
t
)
w
2
p
,
w
2
t
≤
Δ
w
2
+
c
2
(
x
,
t
)
w
1
q
,
x
∈
Ω
,
0
<
t
<
T
1
,
w
1
(
x
,
t
)
≤
∫
Ω
k
1
(
x
,
y
,
t
)
w
1
m
(
y
,
t
)
d
y
,
x
∈
∂
Ω
,
0
<
t
<
T
1
,
w
2
(
x
,
t
)
≤
∫
Ω
k
2
(
x
,
y
,
t
)
w
2
n
(
y
,
t
)
d
y
‍
,
x
∈
∂
Ω
,
0
<
t
<
T
1
,
w
1
(
x
,
0
)
≡
0
,
w
2
(
x
,
0
)
≡
0
,
x
∈
Ω
for some T
1
>
0. By Theorem 13, there exists a unique solution (
h
1
(
x
,
t
)
,
h
2
(
x
,
t
)
) of the following problem:
(49)
h
1
t
=
Δ
h
1
+
c
1
(
x
,
t
)
h
2
p
,
h
2
t
=
Δ
h
2
+
c
2
(
x
,
t
)
h
1
q
,
x
∈
Ω
,
0
<
t
<
T
2
,
h
1
(
x
,
t
)
=
∫
Ω
k
1
(
x
,
y
,
t
)
h
1
m
(
y
,
t
)
d
y
‍
,
x
∈
∂
Ω
,
0
<
t
<
T
2
,
h
2
(
x
,
t
)
=
∫
Ω
k
2
(
x
,
y
,
t
)
h
2
n
(
y
,
t
)
d
y
‍
,
x
∈
∂
Ω
,
0
<
t
<
T
2
,
h
1
(
x
,
0
)
≡
0
,
h
2
(
x
,
0
)
≡
0
,
x
∈
Ω
,
such that h
1
(
x
,
t
)
>
0
,
h
2
(
x
,
t
)
>
0 for x
∈
Ω
¯
,
0
<
t
<
T
2 for some T
2
>
0. Let T
0
=
min
⁡
(
T
1
,
T
2
). In a similar way as in Theorems 13 and 4 we can prove that u
max
⁡
(
x
,
t
)
≥
h
1
(
x
,
t
)
≥
w
1
(
x
,
t
)
,
v
max
⁡
(
x
,
t
)
≥
h
2
(
x
,
t
)
≥
w
2
(
x
,
t
). Put a
1
=
h
1
-
w
1
,
a
2
=
h
2
-
w
2. Now, use an elementary inequality, which is recalled for instance in [20],
(50)
a
k
-
b
k
+
c
k
≥
(
a
-
b
+
c
)
k
,
where 0
<
k
≤
1,  min
⁡
(
a
,
b
,
c
)
>
0, and max
⁡
(
a
,
c
)
≤
b
≤
a
+
c. Then we obtain
(51)
a
1
t
≥
Δ
a
1
+
c
1
(
x
,
t
)
a
2
p
,
a
2
t
≥
Δ
a
2
+
c
2
(
x
,
t
)
a
1
q
,
x
∈
Ω
,
0
<
t
<
T
0
,
a
1
(
x
,
t
)
≥
∫
Ω
k
1
(
x
,
y
,
t
)
a
1
m
(
y
,
t
)
d
y
,
x
∈
∂
Ω
,
0
<
t
<
T
0
,
a
2
(
x
,
t
)
≥
∫
Ω
k
2
(
x
,
y
,
t
)
a
2
n
(
y
,
t
)
d
y
‍
,
x
∈
∂
Ω
,
0
<
t
<
T
0
,
a
1
(
x
,
0
)
≡
0
,
a
2
(
x
,
0
)
≡
0
,
x
∈
Ω
.
We show that a
1
(
x
,
t
)
>
0
,
a
2
(
x
,
t
)
>
0 in Q
T
0. In fact, otherwise, by Theorem 9 there exists t
¯
∈
(
0
,
T
0
) such that a
1
(
x
,
t
)
=
a
2
(
x
,
t
)
≡
0 in Q
t
¯. Suppose that c
1
(
x
0
,
0
)
>
0 for some x
0
∈
Ω. Then we get
(52)
c
1
(
h
2
+
v
)
p
=
c
1
(
w
2
+
v
)
p
=
c
1
v
max
⁡
p
=
(
u
max
⁡
)
t
-
Δ
u
max
⁡
=
(
w
1
+
u
)
t
-
Δ
(
w
1
+
u
)
=
(
h
1
+
u
)
t
-
Δ
(
h
1
+
u
)
=
c
1
(
h
2
p
+
v
p
)
for x
∈
Ω and 0
<
t
<
t
¯. This is a contradiction since h
2
(
x
,
t
)
>
0
,
v
(
x
,
t
)
>
0 in Q
T
0
,
c
1
(
x
0
,
t
0
)
>
0 for some t
0
∈
(
0
,
t
¯
), and 0
<
p
<
1.
Ifk
1
(
x
,
y
0
,
0
)
>
0 for any x
∈
∂
Ω and some y
0
∈
∂
Ω we can obtain a contradiction by another way. Indeed,
(53)
∫
Ω
‍
k
1
(
x
,
y
,
t
)
(
h
1
m
(
y
,
t
)
+
u
m
(
y
,
t
)
)
d
y
=
h
1
(
x
,
t
)
+
u
(
x
,
t
)
=
w
1
(
x
,
t
)
+
u
(
x
,
t
)
=
u
max
⁡
(
x
,
t
)
=
∫
Ω
k
1
(
x
,
y
,
t
)
u
max
⁡
m
(
y
,
t
)
d
y
‍
=
∫
Ω
k
1
(
x
,
y
,
t
)
(
w
1
(
y
,
t
)
+
u
(
y
,
t
)
)
m
d
y
‍
=
∫
Ω
k
1
(
x
,
y
,
t
)
(
h
1
(
y
,
t
)
+
u
(
y
,
t
)
)
m
‍
d
y
for x
∈
∂
Ω and 0
<
t
<
t
¯ and we get again a contradiction since h
1
(
x
,
t
)
>
0
,
u
(
x
,
t
)
>
0 in Q
T
0, and 0
<
m
<
1.
Sincea
1
>
0
,
a
2
>
0 in Q
T
0 by comparison principle with arguments of Theorems 13 and 4 we conclude that a
1
≥
h
1
,
a
2
≥
h
2 in Q
T
0. This implies (46) and completes the proof for the first case.
Now suppose thatmax
⁡
(
p
,
q
,
m
,
n
)
>
1. Assume, for example, that 0
<
p
≤
1
,
q
>
1
,
0
<
m
<
1
,
n
>
1. Then, as in the first case, we introduce the functions w
1
(
x
,
t
)
,
w
2
(
x
,
t
). We use the following relations:
(54)
u
max
⁡
q
(
x
,
t
)
-
u
q
(
x
,
t
)
=
q
θ
1
q
-
1
(
u
max
⁡
(
x
,
t
)
-
u
(
x
,
t
)
)
≤
M
1
(
u
max
⁡
(
x
,
t
)
-
u
(
x
,
t
)
)
,
v
max
⁡
n
(
x
,
t
)
-
v
n
(
x
,
t
)
=
n
θ
2
n
-
1
(
v
max
⁡
(
x
,
t
)
-
v
(
x
,
t
)
)
≤
M
2
(
v
max
⁡
(
x
,
t
)
-
v
(
x
,
t
)
)
,
where θ
1
,
θ
2 are nonnegative continuous functions which are between u
max
⁡
(
x
,
t
)
,
u
(
x
,
t
) and v
max
⁡
(
x
,
t
),  v
(
x
,
t
), respectively, q
θ
1
q
-
1
≤
M
1
,
n
θ
2
n
-
1
≤
M
2 in Q
T. Then we have that a pair of functions (
w
1
(
x
,
t
)
,
w
2
(
x
,
t
)
) satisfies relations
(55)
w
1
t
≤
Δ
w
1
+
c
1
(
x
,
t
)
w
2
p
,
w
2
t
≤
Δ
w
2
+
c
2
(
x
,
t
)
M
1
w
1
,
x
∈
Ω
,
0
<
t
<
T
1
,
w
1
(
x
,
t
)
≤
∫
Ω
k
1
(
x
,
y
,
t
)
w
1
m
(
y
,
t
)
d
y
‍
,
x
∈
∂
Ω
,
0
<
t
<
T
1
,
w
2
(
x
,
t
)
≤
∫
Ω
k
2
(
x
,
y
,
t
)
M
2
w
2
(
y
,
t
)
d
y
,
x
∈
∂
Ω
,
0
<
t
<
T
1
,
w
1
(
x
,
0
)
≡
0
,
w
2
(
x
,
0
)
≡
0
,
x
∈
Ω
.
Further develop the arguments, as in the first case, only forq
=
1,  n
=
1. Using the linearization of terms with powers greater than 1 in the equations and boundary conditions of (1) as above we can prove the theorem for the remaining cases in a similar way.



---

*Source: 523656-2014-02-20.xml*