# Existence of Solutions for a Class of Quasilinear Parabolic Equations with Superlinear Nonlinearities

Authors: Zhong-Xiang Wang; Gao Jia; Xiao-Juan Zhang
Journal: International Journal of Partial Differential Equations
 (2014)

---

## Abstract

Working in a weighted Sobolev space, this paper is devoted to the study of the boundary value problem for the quasilinear parabolic equations with superlinear growth conditions in a domain ofR
N. Some conditions which guarantee the solvability of the problem are given.


---

## Body

## 1. Introduction

In this paper, we deal with the existence of solutions for the quasilinear parabolic problem:(1)
ρ
D
t
u
+
M
~
u
=
λ
1
u
ρ
+
f
x
,
u
ρ
-
G
,
x
,
t
∈
Ω
~
,
u
∈
H
~
Ω
~
,
Γ
,
where Ω
~
=
Ω
×
T, Ω is an open set in R
N
(
N
≥
1
), T
=
(
-
π
,
π
), H
~
(
Ω
~
,
Γ
) is a weighted Sobolev space, λ
1 is the first eigenvalue of L, and M
~ is a singular quasilinear operator defined by
(2)
M
~
u
=
-
∑
i
,
j
=
1
N
‍
D
i
p
i
1
/
2
x
p
j
1
/
2
x
σ
i
1
/
2
u
σ
j
1
/
2
u
b
i
j
x
D
j
u
+
b
0
(
x
)
σ
0
(
u
)
q
u
,
where u
(
x
,
t
)
:
Ω
~
→
R. The nonlinear part f
(
x
,
s
) satisfies the superlinear growth condition
(3)
f
x
,
s
≤
h
0
(
x
)
+
K
s
θ
-
1
,
∀
s
∈
R
,
a
.
e
.
x
∈
Ω
,
where h
0
(
x
)
∈
L
ρ
θ
*, K is a nonnegative constant, 2
<
θ
<
2
N
/
N
-
1, and θ
*
=
θ
/
θ
-
1.There are a number of results concerning solvability of different boundary problems for quasilinear equations (elliptic and parabolic) in which the nonlinearities satisfy sublinear or linear conditions in the weighted Sobolev space, for example, [1–6].In [1], Shapiro established a new weighted compact Sobolev embedding theorem and proved a series of existence problems for weighted quasilinear elliptic equations and parabolic equations.In [2], working in Sobolev space H
p
,
ρ
1
(
Ω
,
Γ
) only for the first eigenvalue, Rumbos and Shapiro on the basis of [3] by using the generalized Landesman-Lazer conditions discussed the existence of the solutions for weighted quasilinear elliptic equations
(4)
P
u
-
λ
1
u
ρ
=
-
a
x
,
u
u
-
ρ
+
g
x
,
u
ρ
+
h
,
x
in
Ω
,
u
=
0
,
x
on
⁡
Ω
,
where
(5)
P
u
=
-
∑
i
,
j
=
1
N
‍
D
i
p
i
1
/
2
p
j
1
/
2
b
i
j
x
D
j
u
+
ρ
(
x
)
c
(
x
)
u
.In [4], Jia and Zhao obtained the existence of a nontrivial solution for a class of singular quasilinear elliptic equations in weighted Sobolev spaces.However, past research results regarding this kind of parabolic equations on superlinearity in the weighted Sobolev space like (1) are very limited. Two notable exceptions are found in [7, 8], where they discuss the periodic solutions for quasilinear parabolic equations when the nonlinearity may grow superlinearly.Our goal here is to extend these results to the case of quasilinear parabolic operators.In fact, (1) is one of the most useful sets which describe the motion of viscous fluid substances. They are widely used in the design of aircrafts and cars, the study of blood flow, the design of power stations, and so forth. Furthermore, coupled with Maxwell’s equations, the Navier-Stokes equations can also be used to model and study magnetohydrodynamics.The main tools applied in our approaches consist of Galerkin method, Brouwer’s theorem, and a new weighted compact Sobolev-type embedding theorem due to Shapiro.This paper is organized as follows. In Section2, we introduce some necessary assumptions and basic results. In Section 3, five fundamental lemmas are established. The subsequent Section 4 contains proofs of the main results.


## 2. Basic Assumptions and Main Theorem

In this section, we introduce some assumptions and give the main results in this paper.LetΩ
⊂
R
N
(
N
≥
1
) be an open (possibly unbounded) set, Γ
⊂
∂
Ω a fixed closed set (Γ may be an empty set), Ω
~
=
Ω
×
T, and T
=
(
-
π
,
π
). We assume throughout this paper that ρ
(
x
), p
i
(
x
)
(
i
=
1,2
,
…
,
N
) are positive C
0
(
Ω
) functions, function q
(
x
) is a nonnegative C
0
(
Ω
) function (q
(
x
) may be zero), and
(6)
∫
Ω
‍
ρ
x
d
x
<
∞
,
∫
Ω
‍
p
i
x
d
x
<
∞
,
i
=
1,2
,
…
,
N
,
∃
K
>
0
,
s
.
t
.
0
≤
q
(
x
)
≤
K
ρ
(
x
)
,
∀
x
∈
Ω
.We definep
=
(
p
1
,
…
,
p
N
), D
i
u
=
∂
u
/
∂
x
i, i
=
1,2
,
…
,
N, and A is a set of real-valued functions defined as
(7)
A
=
u
:
u
∈
C
0
(
Ω
¯
×
R
)
,
u
(
x
,
t
+
2
π
)
=
u
(
x
,
t
)
,
∀
(
x
,
t
)
∈
Ω
¯
×
R
.
We consider the following pre-Hilbert spaces (see [1]):
(8)
C
~
ρ
0
(
Ω
~
)
=
u
∈
C
0
Ω
~
∣
∫
Ω
~
‍
u
x
,
t
2
ρ
x
d
x
d
t
<
∞
,
with inner product 〈
u
,
v
〉
ρ
~
=
∫
Ω
~
‍
u
(
x
,
t
)
v
(
x
,
t
)
ρ
(
x
)
d
x
d
t, and
(9)
C
~
p
,
ρ
1
Ω
~
,
Γ
=
u
∈
A
∩
C
1
Ω
×
R
∣
u
x
,
t
=
0
,
∫
Ω
~
‍
∑
i
=
1
N
‍
D
i
u
2
p
i
+
u
2
+
D
t
u
2
ρ
<
∞
∀
(
x
,
t
)
∈
Γ
×
R
;
∫
Ω
~
‍
∑
i
=
1
N
‍
D
i
u
2
p
i
+
u
2
+
D
t
u
2
ρ
<
∞
,
with inner product
(10)
u
,
v
H
~
=
∫
Ω
~
‍
∑
i
=
1
N
‍
p
i
D
i
u
D
i
v
+
u
v
+
D
t
u
D
t
v
ρ
d
x
d
t
.
We denote L
~
ρ
2
≜
L
ρ
2
(
Ω
~
) which is the Hilbert space obtained from the completion of C
~
ρ
0
(
Ω
~
) with the norm u
ρ
=
(
〈
u
,
u
〉
ρ
~
)
1
/
2, and denote H
~
≜
H
~
(
Ω
~
,
Γ
) which is the Hilbert space obtained from the completion of the space C
~
p
,
ρ
1
(
Ω
~
,
Γ
) with the norm u
H
~
=
〈
u
,
u
〉
H
~
1
/
2. Similarly, we have L
~
p
i
2
(
i
=
1,2
,
…
,
N
) and L
~
q
2.It is assumed throughout the paper thatσ
i
(
u
)
(
i
=
0,1
,
…
,
N
), b
i
j
(
x
)
(
i
,
j
=
1,2
,
…
,
N
), and b
0
(
x
) meet the following assumptions:(σ-1)
σ
i
(
u
): H
~
→
R is weakly sequentially continuous;
(σ-2)
there areη
0
,
η
1
>
0, s.t. η
0
≤
σ
i
(
u
)
≤
η
1, and σ
i
(
u
) is measurable, for u
∈
H
~;
(b-1)
b
0
(
x
)
,
b
i
j
(
x
)
∈
C
0
(
Ω
)
⋂
‍
L
∞
(
Ω
), where b
i
j
(
x
)
=
b
j
i
(
x
), for x
∈
Ω;
(b-2)
b
0
(
x
)
≥
ε
0
>
0, for x
∈
Ω;
(b-3)
there is a constantc
0
>
0, for x
∈
Ω and ξ
∈
R
N, such that
(11)
∑
i
,
j
=
1
N
‍
b
i
j
(
x
)
ξ
i
ξ
j
≥
c
0
ξ
2
,
ξ
2
=
∑
i
=
1
N
‍
ξ
i
2
.Definition 1.
For the quasilinear differential operatorM
~, the two-form is(12)
M
~
u
,
v
=
∑
i
,
j
=
1
N
‍
∫
Ω
~
‍
p
i
1
/
2
p
j
1
/
2
σ
i
1
/
2
u
σ
j
1
/
2
u
b
i
j
D
j
u
D
i
v
+
∫
Ω
~
‍
q
σ
0
(
u
)
b
0
u
v
,
for
u
,
v
∈
H
~
.
Defining
(13)
L
u
=
-
∑
i
=
1
N
‍
D
i
p
i
D
i
u
+
q
u
,
for u
∈
H
p
,
q
,
ρ
Ω
,
Γ (as described in [5]), and
(14)
L
~
u
=
-
∑
i
=
1
N
‍
D
i
(
p
i
D
i
u
)
+
q
u
,
u
∈
H
~
,
the two-form of L is
(15)
L
(
u
,
v
)
=
∑
i
=
1
N
‍
∫
Ω
‍
p
i
D
i
u
D
i
v
+
∫
Ω
‍
u
v
q
,
u
,
v
∈
H
p
,
q
,
ρ
Ω
,
Γ
,
and the two-form of L
~ is
(16)
L
~
(
u
,
v
)
=
∑
i
=
1
N
‍
∫
Ω
~
‍
p
i
D
i
u
D
i
v
+
∫
Ω
~
‍
u
v
q
,
u
,
v
∈
H
~
.Definition 2.
One says thatM
~ is #
H
~-related to L
~ if the following condition holds:
(17)
lim
⁡
u
H
~
→
∞
M
~
(
u
,
v
)
-
L
~
(
u
,
v
)
u
H
~
=
0
,
uniformly for
v
H
~
≤
1
.Definition 3.
(
Ω
~
,
Γ
) is a simple V
L
(
Ω
~
,
Γ
) if(V
L
-
1)
there is a complete orthonormal system{
φ
n
}
n
=
1
∞ in L
ρ
2. Also φ
n
∈
H
p
,
q
,
ρ
∩
C
2 for all n;
(V
L
-
2)
there is a sequence of eigenvalues{
λ
n
}
n
=
1
∞ with 0
<
λ
1
<
λ
2
≤
λ
3
≤
⋯
≤
λ
n
→
∞ such that L
(
φ
n
,
v
)
=
λ
n
φ
n
,
v
ρ for all v
∈
H
p
,
q
,
ρ
(
Ω
,
Γ
). Also φ
1
>
0 in Ω;
(
V
L
-
3
)
Ω
=
Ω
1
×
⋯
×
Ω
N, where Ω
i
⊂
R is an open set for i
=
1
,
…
,
N;
(
V
L
-
4
)
for eachΩ
i, there are positive functions p
i
*
(
x
)
,
ρ
i
*
(
x
)
∈
C
0
(
Ω
i
) satisfying ∫
Ω
i
‍
(
p
i
*
(
x
)
+
ρ
i
*
(
x
)
)
d
x
<
∞, and ρ
(
x
)
=
ρ
1
*
(
x
1
)
⋯
ρ
N
*
(
x
N
), p
i
(
x
)
=
ρ
1
*
(
x
1
)
⋯
ρ
i
-
1
*
(
x
i
-
1
)
p
i
*
(
x
i
)
ρ
i
+
1
*
(
x
i
+
1
)
⋯
ρ
N
*
(
x
N
);
(
V
L
-
5
)
for eachΩ
i, p
i
*
(
x
), and ρ
i
*
(
x
)
(
i
=
1
,
…
,
N
), there exist h
i
∈
C
0
(
Ω
i
)
⋂
‍
L
ρ
i
*
θ
(
Ω
i
) and y
i
(
t
)
∈
C
0
(
T
)
⋂
‍
L
θ
(
T
) for 2
<
θ
<
∞, such that
(18)
v
s
,
t
≤
h
i
s
y
i
t
v
p
i
*
s
,
ρ
i
*
s
,
∀
v
∈
C
1
Ω
i
×
T
,
∀
s
∈
Ω
i
,
t
∈
T
,where
(19)
v
p
i
*
(
s
)
,
ρ
i
*
(
s
)
2
=
∫
Ω
~
i
‍
p
i
*
s
D
s
v
2
+
ρ
i
*
s
v
2
+
D
t
v
2
d
s
d
t
.There are many examples to illustrate the simpleV
L region. One can refer to [1].Remark 4.
We give an example to establish existence results forM
~ and L
~ which meet Definition 2. Taking N
>
1, Ω
⊂
R
N (a bounded open connected set), and Γ
=
∂
Ω, we set ρ
=
q
=
p
i
=
b
0
=
b
i
j
=
1, for i
=
1,2
,
…
,
N, σ
i
(
u
)
=
σ
i
(
u
), and L
~
=
-
∑
i
=
1
N
‍
D
i
i
u
+
u. Then, it is easy to see from [9] that (
Ω
,
Γ
) is a simple V
L region. We set
(20)
σ
i
(
u
)
=
D
i
u
+
D
i
u
1
+
u
2
+
D
u
2
1
/
2
,
σ
0
(
u
)
=
1
+
1
1
+
u
2
+
D
u
2
1
/
2
.
Also, we define
(21)
M
~
=
-
∑
i
=
1
N
‍
D
i
(
σ
i
(
u
)
D
i
u
)
+
σ
0
(
u
)
u
.
We observe with |
D
u
|
2
=
∑
i
=
1
N
‍
|
D
i
u
|
2 that
(22)
M
~
u
,
v
-
L
~
u
,
v
=
∫
Ω
~
‍
1
1
+
u
2
+
D
u
2
u
v
+
∑
i
=
1
N
‍
D
i
u
D
i
v
≤
2
π
Ω
1
/
2
L
~
v
,
v
1
/
2
.
Clearly, M
~ and L
~ meet Definition 2.Remark 5.
IfM
~ (as (12)) meets (
σ
-
1
)
-
(
σ
-
2
), (b-1)–(b-3), then
(23)
M
~
(
v
,
D
t
v
)
=
0
,
∀
v
∈
C
~
p
,
ρ
1
b
=
v
:
v
∈
C
~
p
,
ρ
1
;
D
t
v
∈
C
~
p
,
ρ
1
.f
(
x
,
s
) meets the following conditions:(
f
-
1
)
f
(
x
,
s
) satisfies the Carathéodory conditions;
(
f
-
2
)
(Superlinear condition) there exists θ with 2
<
θ
<
2
N
/
N
-
1 such that |
f
(
x
,
s
)
|
≤
h
0
(
x
)
+
K
|
s
|
θ
-
1, ∀
s
∈
R, a
.
e
.
x
∈
Ω, where h
0
(
x
)
∈
L
ρ
θ
*. K is a nonnegative constant and θ
*
=
θ
/
θ
-
1.Now we state our main results in this paper.Theorem 6.
Assume that(
Ω
~
,
Γ
) is a simple V
L
(
Ω
~
,
Γ
). Let L given by (13) satisfy the conditions of simple V
L
(
Ω
~
,
Γ
), and let M
~ given by (12) satisfy (b-1)–(b-3), (σ-1)-(σ-2) and be #
H
~-related to L
~. Suppose that G
∈
H
~
′ (the dual of H
~) and that (
f
-
1
)-(
f
-
2
) hold. Suppose furthermore there is a nonnegative function h
~
0
(
x
)
∈
L
ρ
θ
* and a constant ε
0
*
>
0 such that
(24)
s
f
(
x
,
s
)
≤
-
ε
0
*
s
2
+
h
~
0
x
s
,
∀
s
∈
R
,
a
.
e
.
x
∈
Ω
.
Then problem (1) has at least one nontrivial weak solution, that is, ∃
u
*
∈
H
~, such that
(25)
D
t
u
*
,
v
ρ
~
+
M
~
u
*
,
v
=
λ
1
u
*
,
v
ρ
~
+
∫
Ω
~
‍
f
x
,
u
*
v
ρ
-
G
v
,
∀
v
∈
H
~
.Remark 7.
Observing that, forN
=
2
,
f
(
x
,
s
)
=
-
g
(
x
)
s
|
s
|
5
/
3
-
ε
0
*
s, where g
(
x
)
∈
C
0
(
Ω
)
∩
L
∞
(
Ω
) is a positive function and g
(
x
) meets (
f
-
2
) and (24).


## 3.  Preliminary Lemmas

In this section, we introduce some lemmas. First, we introduce some notions.If the conditions of simpleV
L
(
Ω
~
,
Γ
) hold, we get
(26)
φ
~
j
k
c
j
=
1
,
k
=
0
∞
,
∞
∪
φ
~
j
k
s
j
=
1
,
k
=
1
∞
,
∞
is a CONS for
L
~
ρ
2
,
where
(27)
i
φ
~
j
k
c
x
,
t
=
1
2
π
φ
j
x
,
k
=
0
,
j
=
1,2
,
…
,
1
π
φ
j
x
cos
⁡
k
t
,
j
,
k
=
1,2
,
…
,
ii
φ
~
j
k
s
x
,
t
=
1
π
φ
j
x
sin
k
t
,
j
,
k
=
1,2
,
…
.
Obviously, both φ
~
j
k
c and φ
~
j
k
s are in H
~. Defining
(28)
L
1
(
u
,
v
)
=
L
~
(
u
,
v
)
+
u
,
v
ρ
~
,
∀
u
,
v
∈
H
~
,
it is clear that L
1
(
u
,
v
) is an inner product on H
~. From (10), (16), and (28), there are K
1
,
K
2
>
0 such that
(29)
K
1
v
H
~
2
≤
L
1
(
v
,
v
)
+
D
t
v
ρ
2
≤
K
2
v
H
~
2
,
∀
v
∈
H
~
.
For v
∈
L
~
ρ
2, setting
(30)
v
^
c
(
j
,
k
)
=
v
,
φ
~
j
k
c
ρ
~
,
v
^
s
(
j
,
k
)
=
v
,
φ
~
j
k
s
ρ
~
,
and from (
V
L
-
2
), (16), and (28), we see that, for v
∈
H
~,
(31)
L
1
(
v
,
φ
~
j
k
s
)
=
(
λ
j
+
1
)
v
^
s
(
j
,
k
)
,
L
1
(
v
,
φ
~
j
k
c
)
=
(
λ
j
+
1
)
v
^
c
(
j
,
k
)
.Lemma 8.
If{
φ
~
j
k
c
}
j
=
1
,
k
=
0
∞
,
∞
∪
{
φ
~
j
k
s
}
j
=
1
,
k
=
1
∞
,
∞ is a CONS for L
ρ
2
(
Ω
~
) defined by (27), setting
(32)
τ
n
(
v
)
=
∑
j
=
1
n
‍
v
^
c
(
j
,
0
)
φ
~
j
0
c
+
∑
j
,
k
=
1
n
‍
(
v
^
c
(
j
,
k
)
φ
~
j
k
c
+
v
^
s
(
j
,
k
)
φ
~
j
k
s
)
,
then
(33)
lim
⁡
n
→
∞
τ
n
(
v
)
-
v
H
~
=
0
,
∀
v
∈
H
~
.Lemma 9.
(i) Ifv
∈
H
~, then
(34)
L
1
v
,
v
+
D
t
v
ρ
2
=
∑
j
=
1
∞
‍
v
^
c
j
,
0
2
(
λ
j
+
1
)
+
∑
j
,
k
=
1
∞
‍
v
^
c
j
,
k
2
+
v
^
s
j
,
k
2
(
λ
j
+
1
+
k
2
)
.
(ii) If v
∈
L
ρ
2
(
Ω
~
) and L
1
v
,
v
+
D
t
v
ρ
2
<
∞, then v
∈
H
~.Lemma 10.
LetL be as Theorem 6 and assume that (
Ω
~
,
Γ
) is a simple V
L
(
Ω
~
,
Γ
). Then H
~ is compactly imbedded in L
ρ
2
(
Ω
~
).For the proofs of Lemmas8–10, one can refer to [5].We define(35)
S
n
=
v
∈
H
~
:
v
=
∑
j
=
1
n
‍
η
j
0
c
φ
~
j
0
c
+
∑
j
,
k
=
1
n
‍
η
j
k
c
φ
~
j
k
c
+
η
j
k
s
φ
~
j
k
s
.Remark 11.
Ifu
n
∈
S
n, then M
~
(
u
n
,
D
t
u
n
)
=
0.Lemma 12.
LetL be given as (13) and suppose that (
Ω
~
,
Γ
) is a simple V
L
(
Ω
~
,
Γ
). Then H
~ is continuously imbedded in L
ρ
θ
(
Ω
~
) for every θ satisfying 2
<
θ
<
2
N
/
N
-
1, that is, ∃
K
θ
>
0, such that
(36)
∫
Ω
~
‍
u
θ
ρ
d
x
d
t
1
/
θ
≤
K
θ
u
H
~
,
∀
u
∈
H
~
.Proof .
To establish the lemma, we need to only prove the case that(37)
∫
Ω
~
‍
v
θ
ρ
d
x
d
t
1
/
θ
≤
K
θ
v
H
~
,
where
v
∈
C
~
p
,
ρ
1
b
Ω
~
,
Γ
,
for
N
≥
2
.
For fixed(
x
2
,
…
,
x
N
)
,
v
(
x
1
,
x
2
,
…
,
x
N
,
t
) is a function only depending on (
x
1
,
t
) in C
1
(
Ω
~
1
). Hence, with (18) and (19), one gets
(38)
v
x
1
,
x
2
,
…
,
x
N
,
t
≤
h
1
(
x
1
)
y
1
(
t
)
g
1
1
/
2
(
x
2
,
…
,
x
N
)
,
where g
1
(
x
2
,
…
,
x
N
)
=
∫
Ω
~
1
‍
[
|
D
1
v
(
x
,
t
)
|
2
p
1
*
(
x
1
)
+
(
|
v
(
x
,
t
)
|
2
+
|
D
t
v
(
x
,
t
)
|
2
)
ρ
1
*
(
x
1
)
]
d
x
1
d
t. Applying the same kind of reasoning to (
x
i
,
t
) for fixed (
x
1
,
…
,
x
i
-
1
,
x
i
+
1
,
…
,
x
N
), we have
(39)
v
x
,
t
N
≤
∏
i
=
1
N
‍
h
i
(
x
i
)
y
i
(
t
)
g
i
1
/
2
(
x
˘
i
)
,
where
(40)
x
˘
i
=
x
1
,
…
,
x
i
-
1
,
x
i
+
1
,
…
,
x
N
,
(41)
g
i
x
˘
i
=
∫
Ω
~
i
‍
D
i
v
x
,
t
2
p
i
*
(
x
i
)
+
(
D
t
v
x
,
t
2
+
v
x
,
t
2
)
ρ
i
*
(
x
i
)
d
x
i
d
t
.
By (39), it follows that
(42)
v
x
,
t
θ
≤
∏
i
=
1
N
‍
y
i
t
θ
/
N
∏
i
=
2
N
‍
h
i
x
i
θ
/
N
g
1
x
˘
1
θ
/
2
N
h
1
x
1
θ
/
N
×
∏
i
=
2
N
‍
g
i
x
˘
i
θ
/
2
N
,
with 2
<
θ
<
2
N
/
N
-
1
(
N
≥
2
). Taking r
1
=
β
=
2
N
/
2
N
-
(
N
-
1
)
θ and r
2
=
⋯
=
r
N
=
γ
=
2
N
/
θ and applying the generalized version of Hölder inequality on Ω
1, we deduce
(43)
∫
Ω
1
‍
v
x
,
t
θ
ρ
1
*
(
x
1
)
d
x
1
≤
∏
i
=
1
N
‍
y
i
t
θ
/
N
∏
i
=
2
N
‍
h
i
x
i
θ
/
N
g
1
x
˘
1
θ
/
2
N
·
∫
Ω
1
‍
h
1
x
1
β
θ
/
N
ρ
1
*
x
1
d
x
1
1
/
β
×
∏
i
=
2
N
‍
∫
Ω
1
‍
g
i
x
˘
i
ρ
1
*
x
1
d
x
1
1
/
γ
.
Integrating the last inequality on both sides with respect tox
2, we get
(44)
∫
Ω
2
‍
∫
Ω
1
‍
v
x
,
t
θ
ρ
1
*
(
x
1
)
ρ
2
*
(
x
2
)
d
x
1
d
x
2
≤
∏
i
=
1
N
‍
y
i
t
θ
/
N
∏
i
=
3
N
‍
h
i
x
i
θ
/
N
∫
Ω
1
‍
g
2
x
˘
2
ρ
1
*
x
1
d
x
1
1
/
γ
·
∏
i
=
1
2
‍
∫
Ω
i
‍
h
i
x
i
β
θ
/
N
ρ
i
*
x
i
d
x
i
1
/
β
×
∫
Ω
2
‍
g
1
x
˘
1
ρ
2
*
x
2
d
x
2
1
/
γ
·
∏
i
=
3
N
‍
∫
Ω
2
‍
∫
Ω
1
‍
g
i
x
˘
i
ρ
1
*
x
1
ρ
2
*
x
2
d
x
1
d
x
2
1
/
γ
.
We continue integrating with respect to x
3
,
…
,
x
N
-
1 and eventually find
(45)
∫
Ω
N
-
1
‍
∫
Ω
N
-
2
‍
⋯
∫
Ω
1
‍
v
x
,
t
θ
ρ
1
*
x
1
⋯
ρ
N
-
2
*
x
N
-
2
×
ρ
N
-
1
*
(
x
N
-
1
)
d
x
1
⋯
d
x
N
-
2
d
x
N
-
1
≤
∏
i
=
1
N
‍
y
i
t
θ
/
N
×
∏
i
=
1
N
-
1
‍
∫
Ω
i
‍
h
i
x
i
β
θ
/
N
ρ
i
*
x
i
d
x
i
1
/
β
h
N
x
N
θ
/
N
·
∫
Ω
N
-
2
‍
⋯
∫
Ω
1
‍
g
N
-
1
x
˘
N
-
1
ρ
1
*
x
1
∫
Ω
1
‍
⋯
ρ
N
-
2
*
x
N
-
2
d
x
1
⋯
d
x
N
-
2
1
/
γ
·
∫
Ω
N
-
1
‍
∫
Ω
N
-
2
‍
⋯
∫
Ω
1
‍
g
N
(
x
˘
N
)
ρ
1
*
(
x
1
)
⋯
ρ
N
-
2
*
(
x
N
-
2
)
×
ρ
N
-
1
*
(
x
N
-
1
)
d
x
1
⋯
d
x
N
-
2
d
x
N
-
1
∫
Ω
1
‍
g
N
1
/
γ
·
∫
Ω
N
-
1
‍
∫
Ω
N
-
3
‍
⋯
∫
Ω
1
‍
g
N
-
2
(
x
˘
N
-
2
)
ρ
1
*
(
x
1
)
⋯
ρ
N
-
3
*
(
x
N
-
3
)
×
ρ
N
-
1
*
x
N
-
1
d
x
1
⋯
d
x
N
-
3
d
x
N
-
1
∫
Ω
1
‍
1
/
γ
⋯
·
∫
Ω
N
-
1
‍
⋯
∫
Ω
3
‍
∫
Ω
1
‍
g
2
x
˘
2
∫
Ω
1
‍
ρ
1
*
x
1
ρ
3
*
x
3
⋯
ρ
N
-
1
*
×
∫
Ω
N
-
1
‍
x
N
-
1
d
x
1
d
x
3
⋯
d
x
N
-
1
1
/
γ
·
∫
Ω
N
-
1
‍
∫
Ω
N
-
2
‍
⋯
∫
Ω
3
‍
∫
Ω
2
‍
g
1
x
˘
1
×
ρ
2
*
(
x
2
)
ρ
3
*
(
x
3
)
⋯
ρ
N
-
2
*
(
x
N
-
2
)
×
ρ
N
-
1
*
(
x
N
-
1
)
d
x
2
d
x
3
∫
Ω
N
-
1
‍
⋯
d
x
N
-
2
d
x
N
-
1
1
/
γ
.
Furthermore, we have
(46)
∫
Ω
~
N
‍
∫
Ω
N
-
1
‍
∫
Ω
N
-
2
‍
⋯
∫
Ω
1
‍
v
x
,
t
θ
ρ
1
*
x
1
⋯
ρ
N
-
2
*
x
N
-
2
×
ρ
N
-
1
*
x
N
-
1
ρ
N
*
x
N
d
x
1
⋯
d
x
N
-
2
d
x
N
-
1
d
x
N
d
t
≤
∫
T
‍
∏
i
=
1
N
‍
y
i
t
θ
/
N
d
t
×
∏
i
=
1
N
‍
∫
Ω
i
‍
h
i
x
i
β
θ
/
N
ρ
i
*
x
i
d
x
i
1
/
β
∏
i
=
1
N
‍
J
i
1
/
γ
with
(47)
J
i
=
∫
Ω
N
‍
⋯
∫
Ω
i
+
1
‍
∫
Ω
i
-
1
‍
⋯
∫
Ω
1
‍
g
i
x
˘
i
ρ
1
*
x
1
⋯
ρ
i
-
1
*
x
i
-
1
ρ
i
+
1
*
(
x
i
+
1
)
⋯
ρ
N
*
(
x
N
)
d
x
1
⋯
d
x
i
-
1
d
x
i
+
1
⋯
d
x
N
.
From (18) we see that h
i
∈
C
0
(
Ω
i
)
∩
L
ρ
i
*
θ
(
Ω
i
) and y
i
(
t
)
∈
C
0
(
T
)
∩
L
θ
(
T
). Hence, we obtain from (46) that there is a constant K
θ such that
(48)
∫
Ω
~
‍
v
x
,
t
θ
ρ
x
d
x
d
t
1
/
θ
≤
K
θ
∏
i
=
1
N
‍
J
i
1
/
2
N
.
By (41), (47), and p
i
(
x
)
=
ρ
1
*
(
x
1
)
⋯
ρ
i
-
1
*
(
x
i
-
1
)
p
i
*
(
x
i
)
ρ
i
+
1
*
(
x
i
+
1
)
⋯
ρ
N
*
(
x
N
), we see that
(49)
J
i
=
∫
Ω
~
‍
D
i
v
x
,
t
2
p
i
x
+
D
t
v
x
,
t
2
+
v
x
,
t
2
ρ
x
d
x
d
t
.
Therefore J
i
≤
v
H
~
2 for i
=
1
,
…
,
N, and ∏
i
=
1
N
‍
J
i
≤
v
H
~
2
N.
By using (48), we get (36). The proof of Lemma 12 is complete.Lemma 13.
LetL
u be given by (13) and suppose that (
Ω
~
,
Γ
) is a simple V
L
(
Ω
~
,
Γ
). Then, for N
≥
2, H
~ is compactly imbedded in L
ρ
θ
(
Ω
~
) for 2
<
θ
<
2
N
/
N
-
1. For N
=
1, H
~ is compactly imbedded in L
ρ
θ
(
Ω
~
) for 2
<
θ
<
∞.Lemma13 is an immediate byproduct of [1, Theorem 9].


## 4. Proof of Theorem6

In this section, we will give the proof of Theorem6. In order to do this, we divide the proof into three parts.Lemma 14.
Let all the assumptions in Theorem6 hold. Then, for n
≥
2, ∃
u
n
∈
S
n such that
(50)
D
t
u
n
,
v
ρ
~
+
M
~
u
n
,
v
=
λ
1
-
1
n
u
n
,
v
ρ
~
+
f
(
x
,
u
n
)
,
v
ρ
~
-
G
(
v
)
,
∀
v
∈
S
n
.Proof.
To prove the lemma, we first observe from (35) that
(51)
(
i
)
v
∈
S
n
⟹
D
t
v
∈
S
n
,
ii
D
t
α
φ
~
j
k
c
+
β
φ
~
j
k
s
,
α
φ
~
j
k
c
+
β
φ
~
j
k
s
ρ
~
=
0
,
for
j
,
k
≥
1
,
α
,
β
∈
R
.
Letψ
i
i
=
1
2
n
2
+
n be an enumeration of {
φ
~
j
k
c
}
j
=
1
,
k
=
0
n
,
n
∪
{
φ
~
j
k
s
}
j
=
1
,
k
=
1
n
,
n, and set
(52)
n
*
=
(
j
0
+
j
1
-
1
)
(
2
n
+
1
)
.
So {
ψ
i
}
i
=
1
n
* is an enumeration of {
φ
~
j
k
c
}
j
=
1
,
k
=
0
j
0
+
j
1
-
1
,
n
∪
{
φ
~
j
k
s
}
j
=
1
,
k
=
1
j
0
+
j
1
-
1
,
n, where n
≥
n
0.
Forα
=
α
1
,
…
,
α
2
n
2
+
n, setting
(53)
u
=
∑
i
=
1
2
n
2
+
n
‍
α
i
ψ
i
,
u
~
=
∑
i
=
1
2
n
2
+
n
‍
δ
i
α
i
ψ
i
,
where
(54)
δ
i
=
-
1
,
1
≤
i
≤
n
*
,
1
,
n
*
+
1
≤
i
≤
2
n
2
+
n
,
from (52)–(54), we have
(55)
u
~
=
∑
j
=
1
n
‍
δ
j
u
^
c
(
j
,
0
)
φ
~
j
0
c
+
∑
j
,
k
=
1
n
‍
δ
j
[
u
^
c
(
j
,
k
)
φ
~
j
k
c
+
u
^
s
(
j
,
k
)
φ
~
j
k
s
]
,
u
=
∑
j
=
1
n
‍
u
^
c
(
j
,
0
)
φ
~
j
0
c
+
∑
j
,
k
=
1
n
‍
[
u
^
c
(
j
,
k
)
φ
~
j
k
c
+
u
^
s
(
j
,
k
)
φ
~
j
k
s
]
,
δ
j
=
-
1
,
1
≤
j
≤
j
0
+
j
1
-
1
,
1
,
j
0
+
j
1
≤
j
≤
n
,
L
1
u
,
u
+
D
t
u
ρ
2
=
∑
j
=
1
n
‍
u
^
c
j
,
0
2
(
λ
j
+
1
)
+
∑
j
=
1
n
‍
∑
k
=
1
n
‍
u
^
c
j
,
k
2
+
u
^
s
j
,
k
2
(
λ
j
+
1
+
k
2
)
.
Further, (29) yields
(56)
K
1
u
H
~
2
≤
L
1
(
u
,
u
)
+
D
t
u
ρ
2
≤
∑
j
=
1
n
‍
u
^
c
j
,
0
2
(
λ
n
+
1
)
+
∑
j
,
k
=
1
n
‍
[
u
^
c
j
,
k
2
+
u
^
s
j
,
k
2
]
(
λ
n
+
1
+
n
2
)
≤
(
λ
n
+
1
+
n
2
)
u
ρ
2
,
∀
u
∈
S
n
.
For m
≥
2, we set
(57)
f
m
(
x
,
s
)
=
f
x
,
m
,
for
m
≤
s
,
f
x
,
s
,
for
-
m
≤
s
≤
m
,
f
x
,
m
,
for
s
≤
-
m
,
and we claim that ∃
u
m
*
∈
S
n such that
(58)
D
t
u
m
*
,
v
ρ
~
+
M
~
u
m
*
,
v
=
λ
1
-
1
n
u
m
*
,
v
ρ
~
+
∫
Ω
~
‍
f
m
(
x
,
u
m
*
)
v
ρ
-
G
(
v
)
,
∀
v
∈
S
n
.
By (f-2) and the above,
(59)
f
m
x
,
s
≤
h
0
x
+
K
m
θ
-
1
,
∀
s
∈
R
,
a
.
e
.
x
∈
Ω
,
where h
0
(
x
)
∈
L
ρ
θ
* and θ
*
=
θ
/
θ
-
1. Therefore, f
m
(
x
,
u
)
∈
L
ρ
θ
*
(
Ω
~
) and ∫
Ω
~
‍
f
m
x
,
u
v
ρ is well defined for v
∈
S
n by Lemma 12.
To establish (58), we set
(60)
F
i
α
=
D
t
u
,
ψ
i
ρ
~
+
M
~
(
u
,
ψ
i
)
-
λ
1
-
1
n
u
,
ψ
i
ρ
~
-
f
m
x
,
u
,
ψ
i
ρ
~
+
G
(
ψ
i
)
for i
=
1
,
…
,
n, and we observe from (51) that
(61)
∑
i
=
1
2
n
2
+
n
‍
F
i
(
α
)
α
i
=
I
(
α
)
+
I
I
(
α
)
,
where
(62)
I
(
α
)
=
L
~
(
u
,
u
)
-
λ
1
-
1
n
u
,
u
ρ
~
,
(63)
I
I
(
α
)
=
M
~
(
u
,
u
)
-
L
~
(
u
,
u
)
-
∫
Ω
~
‍
f
m
(
x
,
u
)
u
ρ
+
G
(
u
)
.
Now, from (62),
(64)
I
(
α
)
=
∑
i
=
1
2
n
2
+
n
‍
(
λ
i
-
λ
1
)
α
i
2
+
α
2
n
,
and according to Lemma 12, (56), (59), (63), and the fact that M
~ is #
H
~-related to L
~,
(65)
lim
⁡
|
α
|
→
∞
I
I
α
α
2
=
0
.
We see from (61) that ∃
s
0
>
0 such that
(66)
∑
i
=
1
2
n
2
+
n
‍
F
i
(
α
)
α
i
≥
α
2
2
n
,
for
α
>
s
0
.
Therefore, by virtue of generalized Brouwer’s theorem [10], there exists α
*
=
(
α
1
*
,
…
,
α
2
n
2
+
n
*
) such that F
i
(
α
*
)
=
0, for i
=
1
,
…
,
n. We set u
m
*
=
∑
i
=
1
2
n
2
+
n
‍
α
i
*
ψ
i and see from (60) that (58) does indeed hold.
By the definition off
m
(
x
,
s
) and (24), for m
≥
2,
(67)
s
f
m
x
,
s
≤
h
~
0
x
s
,
∀
s
∈
R
,
a
.
e
.
x
∈
Ω
,
where h
~
0
(
x
)
∈
L
ρ
θ
*
(
Ω
).
We claim(68)
∃
K
1
>
0
s
.
t
.
u
m
*
ρ
≤
K
1
,
∀
m
>
2
.
Suppose that (68) is false. Then there is a subsequence (which for ease of notation, we take to be the full sequence) such that
(69)
lim
⁡
|
m
|
→
∞
u
m
*
ρ
=
∞
.
Insertingu
m
* in place of v in (58), we find
(70)
L
~
(
u
m
*
,
u
m
*
)
-
λ
1
-
1
n
u
m
*
,
u
m
*
ρ
~
=
∫
Ω
~
‍
f
m
(
x
,
u
m
*
)
u
m
*
ρ
-
G
(
u
m
*
)
+
L
~
(
u
m
*
,
u
m
*
)
-
M
~
(
u
m
*
,
u
m
*
)
,
and the left-hand side of (70) is
(71)
∑
j
=
1
n
‍
u
^
m
*
c
j
,
0
2
(
λ
j
-
λ
1
)
+
∑
j
,
k
=
1
n
‍
(
u
^
m
*
c
j
,
k
2
+
u
^
m
*
s
j
,
k
2
)
(
λ
j
-
λ
1
)
+
u
m
*
ρ
2
n
.
So we see from (67) and Hölder inequality that
(72)
u
m
*
ρ
2
n
≤
h
~
0
L
ρ
θ
*
u
m
*
L
ρ
θ
+
G
u
m
*
+
L
~
u
m
*
,
u
m
*
-
M
~
u
m
*
,
u
m
*
.
From (36), (56), (69), and the fact that M
~ is #
H
~-related to L
~, dividing both sides of (72) by u
m
*
ρ
2 and taking the limit as m
→
∞, we obtain that 1
/
n
≤
0. But n is a positive integer. So we get a contradiction. Therefore, (68) does indeed hold.
Hence, from (56), (68), and u
m
*
∈
S
n, for m
≥
2, there is a subsequence {
u
m
j
*
}
j
=
1
∞ and a u
n
∈
S
n [11] such that
(73)
lim
⁡
j
→
∞
u
m
j
*
-
u
n
H
~
=
0
;
lim
⁡
j
→
∞
u
m
j
*
(
x
,
t
)
=
u
n
(
x
,
t
)
,
a
.
e
.
(
x
,
t
)
∈
Ω
~
;
lim
⁡
j
→
∞
D
i
u
m
j
*
x
,
t
=
D
i
u
n
x
,
t
,
a
.
e
.
x
,
t
∈
Ω
~
,
i
=
1
,
…
,
N
.
It is easy to check thatM
~
(
u
m
j
*
,
v
)
→
M
~
(
u
n
,
v
) as j
→
∞, ∀
v
∈
S
n, and from Lemma 12 we obtain that lim
⁡
j
→
∞
∫
Ω
~
‍
|
u
m
j
*
-
u
n
|
θ
ρ
=
0. Consequently, there exist a function W
∈
L
ρ
θ
(
Ω
~
) and a subsequence {
u
m
j
k
*
}
k
=
1
∞ such that
(74)
u
m
j
k
*
x
,
t
≤
W
(
x
,
t
)
a
.
e
.
(
x
,
t
)
∈
Ω
~
,
∀
k
,
lim
⁡
k
→
∞
f
m
j
k
(
x
,
u
m
j
k
*
)
=
f
(
x
,
u
n
)
a
.
e
.
(
x
,
t
)
∈
Ω
~
.
By using (f-2), Hölder inequality, and the Lebesgue dominated convergence theorem, we conclude that
(75)
lim
⁡
k
→
∞
∫
Ω
~
‍
f
m
j
k
(
x
,
u
m
j
k
*
)
v
ρ
=
∫
Ω
~
‍
f
(
x
,
u
n
)
v
ρ
,
∀
v
∈
S
n
.
Hence replacing u
m
* by u
m
j
k
* in (58) and passing to the limit as k
→
∞, we consequently obtain that
(76)
D
t
u
n
,
v
ρ
~
+
M
~
u
n
,
v
=
λ
1
-
1
n
u
n
,
v
ρ
~
+
f
(
x
,
u
n
)
,
v
ρ
~
-
G
(
v
)
,
∀
v
∈
S
n
,
and the proof of the lemma is complete.Lemma 15.
Let the conditions in Lemma14 hold. Then the sequence {
u
n
} obtained in Lemma 14 is uniformly bounded in H
~ with respect to the norm u
n
H
~
=
〈
u
n
,
u
n
〉
H
~
1
/
2.Proof .
According to Lemma14, we obtain a sequence u
n
n
=
2
∞, where
(77)
u
n
=
∑
i
=
1
2
n
2
+
n
‍
α
(
i
,
n
)
ψ
i
,
and u
n satisfies
(78)
D
t
u
n
,
v
ρ
~
+
M
~
u
n
,
v
=
λ
1
-
1
n
u
n
,
v
ρ
~
+
f
(
x
,
u
n
)
,
v
ρ
~
-
G
(
v
)
,
∀
v
∈
S
n
.
We claim there is a constantK such that
(79)
u
n
H
~
≤
K
,
∀
n
.
Suppose that the assertion is false. Then, without loss of generality, we assume
(80)
lim
⁡
n
→
∞
u
n
H
~
=
∞
.
Puttingu
n in place of v in (78), we have
(81)
D
t
u
n
,
u
n
ρ
~
+
M
~
u
n
,
u
n
=
λ
1
-
1
n
u
n
,
u
n
ρ
~
+
f
(
x
,
u
n
)
,
u
n
ρ
~
-
G
(
u
n
)
.
Next, we observe from (77) and Lemma 8 that
(82)
L
~
u
n
,
u
n
=
∑
j
=
1
n
‍
u
^
n
c
j
,
0
2
λ
j
+
∑
j
,
k
=
1
n
‍
(
u
^
n
c
j
,
k
2
+
u
^
n
s
j
,
k
2
)
λ
j
≥
λ
1
u
n
ρ
2
and from (78), (24), and Hölder inequality that
(83)
ε
0
*
u
n
ρ
2
≤
∥
h
~
0
∥
L
~
ρ
θ
*
∥
u
n
∥
L
~
ρ
θ
+
G
u
n
+
L
~
u
n
u
n
-
M
~
u
n
,
u
n
.
Hence it follows from Lemma12, G
∈
H
~
′, M
~ being #
H
~-related to L
~, ε
0
* being strictly positive, and (83) that
(84)
lim
⁡
n
→
∞
u
n
ρ
u
n
H
~
=
0
.
Next, we observe from (12) and (16) that there is a positive constant c such that
(85)
c
L
~
(
u
n
,
u
n
)
≤
M
~
(
u
n
,
u
n
)
,
∀
n
.
Putting D
t
u
n for v in (78), we see that there exists K
>
0 such that
(86)
D
t
u
n
ρ
≤
K
.
But then it follows from (81), (24), and D
t
u
n
ρ
≤
K that
(87)
c
L
~
u
n
,
u
n
+
ε
0
*
u
n
ρ
2
+
D
t
u
n
ρ
2
≤
h
~
0
L
~
ρ
θ
*
u
n
L
~
ρ
θ
+
λ
1
u
n
ρ
2
+
G
u
n
+
K
.
So, we see from (28), (29), and (87) that there is a positive constant K
1 such that
(88)
u
n
H
~
2
≤
K
1
(
∥
h
~
0
∥
L
~
ρ
θ
*
∥
u
n
∥
L
~
ρ
θ
+
λ
1
u
n
ρ
2
+
G
u
n
+
K
)
.
Consequently, dividing both sides of (88) by u
n
H
~
2 and passing to the limit as n
→
∞, from Lemma 12, the fact that G
∈
H
~
′, (80), and (84), we obtain that 1
≤
0. This is a contradiction. Hence (80) does not hold and (79) is true.Proof of Theorem6.
SinceH
~ is a separable Hilbert space, we see from (79) and Lemmas 9 and 13 that there exist a subsequence (for the sake of simplicity, we take to be a full sequence) and a function u
*
∈
H
~ with the following properties [11]:
(89)
1
lim
⁡
n
→
∞
u
n
-
u
*
ρ
+
∫
Ω
~
‍
u
n
-
u
*
θ
ρ
=
0
;
2
∃
w
x
,
t
∈
L
~
ρ
2
∩
L
~
ρ
θ
,
s
.
t
.
u
n
x
,
t
≤
w
x
,
t
,
a
.
e
.
x
,
t
∈
Ω
~
,
∀
n
;
3
lim
⁡
n
→
∞
u
n
x
,
t
=
u
*
x
,
t
,
a
.
e
.
x
,
t
∈
Ω
~
;
4
lim
⁡
n
→
∞
D
i
u
n
,
v
p
i
~
=
D
i
u
*
,
v
p
i
~
,
∀
v
∈
L
~
p
i
2
,
i
=
1
,
…
,
N
;
5
lim
⁡
n
→
∞
u
n
,
v
q
~
=
u
*
,
v
q
~
,
∀
v
∈
L
~
q
2
;
6
lim
⁡
n
→
∞
G
u
n
=
G
u
*
;
(
7
)
lim
⁡
n
→
∞
D
t
u
n
,
v
ρ
~
=
D
t
u
*
,
v
ρ
~
,
∀
v
∈
L
~
ρ
2
.
We letv
J
∈
S
J. Then it follows from (78) that, for n
≥
J,
(90)
D
t
u
n
,
v
J
ρ
~
+
M
~
u
n
,
v
J
=
λ
1
-
1
n
u
n
,
v
J
ρ
~
+
f
(
x
,
u
n
)
,
v
J
ρ
~
-
G
(
v
J
)
.
Now, from (
σ
-
1
)
-
(
σ
-
2
) and (89)(
1
)
(
4
)
(
7
), we obtin that
(91)
lim
⁡
n
→
∞
D
t
u
n
,
v
J
ρ
~
+
M
~
(
u
n
,
v
J
)
=
D
t
u
n
,
v
J
ρ
~
+
M
~
u
n
,
v
J
.
Next, we observe from (f-2) and (89)(2) that
(92)
f
x
,
u
n
≤
K
w
x
,
t
θ
-
1
+
h
0
x
a
.
e
.
x
,
t
∈
Ω
~
,
n
=
1,2
,
…
,
where w
∈
L
~
ρ
θ and h
0
∈
L
ρ
θ
*. From (f-1) and (89)(3), we further obtain that
(93)
lim
⁡
n
→
∞
f
(
x
,
u
n
(
x
,
t
)
)
=
f
(
x
,
u
*
(
x
,
t
)
)
a
.
e
.
(
x
,
t
)
∈
Ω
~
.
Note that |
w
|
θ
-
1
∈
L
~
ρ
θ
*. Hence we conclude from Hölder inequality, the Lebesgue dominated convergence theorem, Lemma 12, and (92) that
(94)
lim
⁡
n
→
∞
∫
Ω
~
‍
f
(
x
,
u
n
(
x
,
t
)
)
v
J
ρ
=
∫
Ω
~
‍
f
(
x
,
u
*
(
x
,
t
)
)
v
J
ρ
.
Then passing to the limit on both sides of (90), we obtain from (91), (94), and (89)(
1
) that
(95)
D
t
u
*
,
v
J
ρ
+
M
~
u
*
,
v
J
=
λ
1
u
*
,
v
J
ρ
+
∫
Ω
~
‍
f
(
x
,
u
*
(
x
,
t
)
)
v
J
ρ
-
G
(
v
J
)
.
Next, for given v
∈
H
~, we replace v
J with τ
J
(
v
) (defined as (32)) in (95). From (29) and Lemma 8, τ
J
(
v
)
-
v
H
~
→
0 as J
→
∞. From (95) and Lemma 12, it is then an easy matter to obtain that
(96)
D
t
u
*
,
v
ρ
~
+
M
~
u
*
,
v
=
λ
1
u
*
,
v
ρ
~
+
f
x
,
u
*
,
v
ρ
~
-
G
(
v
)
,
∀
v
∈
H
~
,
and the proof of Theorem 6 is established.



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*Source: 436369-2014-12-21.xml*