# On an Anisotropic Parabolic Equation on the Domain with a Disjoint Boundary

Authors: Huashui Zhan
Journal: Journal of Function Spaces
 (2018)

---

## Abstract

Consider the anisotropic parabolic equation with the variable exponentsv
t
=
∑
i
=
1
n
(
b
i
(
x
)
v
x
i
q
i
x
-
2
v
x
i
)
x
i
, where b
i
(
x
)
,
q
i
(
x
)
∈
C
1
(
Ω
¯
), q
i
(
x
)
>
1, and b
i
(
x
)
≥
0. If {
b
i
(
x
)
} is not degenerate on Σ
p
⊂
∂
Ω, a part of the boundary, but is degenerate on the remained part ∂
Ω
∖
Σ
p, then the boundary value condition is imposed on Σ
p, but there is no boundary value condition required on ∂
Ω
∖
Σ
p. The stability of the weak solutions can be proved based on the partial boundary value condition v
x
∈
Σ
p
=
0.


---

## Body

## 1. Introduction

Recently, we had considered the anisotropic parabolic equation(1)
v
t
=
∑
i
=
1
n
b
i
x
v
x
i
q
i
x
-
2
v
x
i
x
i
,
x
,
t
∈
Q
T
=
Ω
×
0
,
T
,with the initial-boundary value conditions (2)
v
x
,
0
=
v
0
x
,
x
∈
Ω
,
(3)
v
x
,
t
=
0
,
x
,
t
∈
∂
Ω
×
0
,
T
,where b
i
(
x
)
,
q
i
(
x
)
∈
C
1
(
Ω
¯
), q
i
(
x
)
>
1, b
i
(
x
)
≥
0, and Ω is a bounded domain with a smooth boundary ∂
Ω. If b
i
(
x
)
≡
1, the existence of the weak solution had been proved by Antontsev and Shmarev [1]. If some of b
i
(
x
) are degenerate on the boundary, Zhan [2] had conjectured that, instead of the usual boundary value condition (3), only a partial boundary value condition, (4)
v
x
,
t
=
0
,
x
,
t
∈
Σ
p
×
0
,
T
,should be imposed, while Σ
p is relatively open in ∂
Ω. For example, if (5)
Ω
=
x
1
,
x
2
,
…
,
x
n
:
0
<
x
1
<
1,0
<
x
2
<
1
,
…
,
0
<
x
n
<
1and b
i
(
x
) has some special restrictions, then the explicit Σ
p is given, and the stability of the weak solutions is proved dependent on the partial boundary value condition [2]. However, for a general bounded domain Ω, how to depict out the explicit Σ
p seems very difficult. In this short paper, we give an original attempt, we first assume that (6)
∂
Ω
=
Σ
p
∪
Σ
p
′
,
Σ
p
¯
∩
Σ
p
′
¯
=
∅
,
(7)
b
i
x
≥
c
i
>
0
,
x
∈
Σ
p
,
(8)
b
i
x
=
0
,
x
∈
Σ
p
′
,where c
i is a suitable positive constant, and denote that (9)
q
0
=
m
i
n
x
∈
Ω
¯
q
1
x
,
q
2
x
,
…
,
q
n
-
1
x
,
q
n
x
,
q
0
>
1
,
q
0
=
m
a
x
x
∈
Ω
¯
q
1
x
,
q
2
x
,
…
,
q
n
-
1
x
,
q
n
x
.A simple example satisfies (6) is (10)
Ω
=
x
∈
R
n
:
1
<
x
1
2
+
x
2
2
+
⋯
+
x
n
2
<
4
.Definition 1.
If a functionv
(
x
,
t
) satisfies that (11)
v
∈
L
∞
Q
T
,
∂
v
∂
t
∈
L
2
Q
T
,
b
i
x
v
x
i
q
i
x
∈
L
2
0
,
T
;
L
1
Ω
,for any function φ
,
φ
x
i
∈
L
2
(
0
,
T
;
L
q
i
(
x
)
(
Ω
)
) and φ
x
∈
∂
Ω
=
0, (12)
∬
Q
T
∂
v
∂
t
φ
+
∑
i
=
1
n
b
i
x
v
x
i
q
i
x
-
2
v
x
i
φ
x
i
d
x
d
t
=
0
,
(13)
l
i
m
t
→
0
∫
Ω
u
x
,
t
-
u
0
x
d
x
=
0
,then we say v
(
x
,
t
) is a weak solution of (1) with the initial value condition (2). Besides, if the partial boundary value condition (4) is satisfied in the sense of the trace, then we say that v
(
x
,
t
) is a weak solution of the initial-boundary value problem (1)-(2)-(4).
The existence of the weak solution can be proved by the usual parabolically regularized method [2]. We are not ready to discuss the existence again in this paper. We mainly pay attentions on the stability.Theorem 2.
IfΩ satisfies (6), b
i satisfies (7)-(8), and for large enough m, (14)
m
∫
Ω
∖
Ω
m
b
i
x
∏
j
=
1
n
b
j
x
x
i
q
i
x
d
x
1
/
q
i
+
≤
c
,
v
(
x
,
t
) and u
(
x
,
t
) are two solutions of (1) with the same partial boundary value condition (15)
v
x
,
t
=
u
x
,
t
=
0
,
x
,
t
∈
Σ
p
×
0
,
T
,then (16)
∫
Ω
v
x
,
t
-
u
x
,
t
d
x
≤
∫
Ω
v
x
,
0
-
u
x
,
0
d
x
,where Ω
m
=
{
x
∈
Ω
:
∏
i
=
1
n
b
i
(
x
)
>
1
/
m
}.Remark 3.
Since the domainΩ satisfies (6) and b
i satisfies (7)-(8) and when x is near to Σ
p, (1) is not degenerate, by (14), (17)
∫
Ω
1
δ
∇
v
q
0
d
x
<
∞
,then we can define the trace of v
(
x
,
t
) on Σ
p, and condition (15) is reasonable. Here δ
>
0 is a small enough constant, (18)
Ω
1
δ
=
x
∈
Ω
:
d
x
=
d
i
s
t
x
,
Σ
p
<
δ
.
However, whenΣ
p
¯
∩
Σ
p
′
¯
≠
∅, then (17) is not clear. In this case, only if (19)
∫
Ω
b
i
x
-
1
/
q
i
x
-
1
d
x
<
∞
,then we have a similar conclusion. This is the following theorem.Theorem 4.
If the domainΩ satisfies (20)
∂
Ω
=
Σ
p
∪
Σ
p
′
,
Σ
p
∩
Σ
p
′
=
∅
,
Σ
p
¯
∩
Σ
p
′
¯
≠
∅
,
b
i satisfies (19) and (21)
b
i
x
>
0
,
x
∈
Σ
p
,
b
i
x
=
0
,
x
∈
Σ
p
′
,condition (14) is true when m is large enough, and if v
(
x
,
t
) and u
(
x
,
t
) are two solutions of (1) with the same partial boundary value condition (15), then stability (16) is true.Let us give an example of the domainΩ and b
i
(
x
) in Theorem 4. For example, n
=
2, (22)
Ω
=
D
1
=
x
∈
R
2
:
x
1
2
+
x
2
2
<
1
,
Σ
p
=
x
∈
R
2
:
x
1
2
+
x
2
2
=
1
,
x
2
>
0
,
Σ
p
′
=
x
∈
R
2
:
x
1
2
+
x
2
2
=
1
,
x
2
≤
0
.
(23)
v
t
=
∂
∂
x
1
b
1
x
v
x
1
q
1
x
-
2
v
x
1
+
∂
∂
x
2
b
2
x
v
x
1
q
2
x
-
2
v
x
2
,(24)
b
1
x
=
b
2
x
=
x
2
+
1
-
x
1
2
.At the end of this section, we would like to give a simple comment on the research background of this paper. Equation (1) is the generalized equation of the following equation: (25)
v
t
=
d
i
v
b
x
∇
v
q
x
-
2
∇
v
,
x
,
t
∈
Q
T
,which originally comes from the electrorheological fluids theory (see [3, 4]). If b
(
x
)
≡
1, there are many related papers; one can see [5–7] and the references therein. If b
(
x
)
>
0 when x
∈
Ω but b
x
x
∈
∂
Ω
=
0, then the stability of the weak solutions without the boundary value condition had been studied by Zhan et al. [8–10], provided that the diffusion coefficient b
(
x
) satisfies some other restrictions.


## 2. The Stability

The concepts of the exponent variable spaces,(
L
q
x
Ω, ·
L
q
(
x
)
(
Ω
)
), (
W
1
,
q
(
x
)
(
Ω
)
,
·
W
1
,
q
(
x
)
(
Ω
)
), and W
0
1
,
q
(
x
)
(
Ω
), can be found in [11–15].Lemma 5 (see [11–13]).
Ifp
(
x
) and q
(
x
) are real functions with 1
/
p
(
x
)
+
1
/
q
(
x
)
=
1 and q
(
x
)
>
1, then, for any v
∈
L
p
(
x
)
(
Ω
) and u
∈
L
q
(
x
)
(
Ω
), one has (26)
∫
Ω
v
u
d
x
≤
2
v
L
p
x
Ω
u
L
q
x
Ω
. Moreover, (27)
i
f
v
L
q
x
Ω
=
1
,
t
h
e
n
∫
Ω
v
q
x
d
x
=
1
,
i
f
v
L
q
x
Ω
>
1
,
t
h
e
n
v
L
q
x
q
-
≤
∫
Ω
v
q
x
d
x
≤
v
L
q
x
q
+
,
i
f
v
L
q
x
Ω
<
1
,
t
h
e
n
v
L
q
x
q
+
≤
∫
Ω
v
q
x
d
x
≤
v
L
q
x
q
-
.
One letsk
m
(
s
) be an odd function, and (28)
k
m
s
=
1
,
s
>
1
m
,
m
2
s
2
e
1
-
m
2
s
2
,
0
≤
s
⩽
1
m
.Then, (29)
l
i
m
m
→
∞
k
m
s
=
s
g
n
s
,
s
∈
-
∞
,
+
∞
.
Letφ
(
x
) be a C
1
(
Ω
¯
) function satisfying (30)
φ
x
x
∈
Σ
p
′
=
0
,
φ
x
x
∈
Ω
¯
∖
Σ
p
′
>
0
,
Ω
m
=
x
∈
Ω
:
φ
x
>
1
m
.Theorem 6.
IfΩ satisfies (6), b
i satisfies (7)-(8), and for large enough m, (31)
m
∫
Ω
∖
Ω
m
b
i
x
φ
x
x
i
q
i
x
d
x
1
/
q
i
+
≤
c
,
v
(
x
,
t
) and u
(
x
,
t
) are two solutions of (1) with the same partial boundary value condition(32)
v
x
,
t
=
u
x
,
t
=
0
,
x
,
t
∈
Σ
p
×
0
,
T
,then(33)
∫
Ω
v
x
,
t
-
u
x
,
t
d
x
≤
∫
Ω
v
x
,
0
-
u
x
,
0
d
x
.Proof.
We letv
(
x
,
t
) and u
(
x
,
t
) be two weak solutions of (1) with the partial boundary value condition (32). Then, k
m
v
-
u
x
∈
Σ
p
=
0.
Let(34)
φ
m
x
=
1
,
if
x
∈
Ω
m
,
m
φ
x
,
if
x
∈
Ω
∖
Ω
m
.Then φ
m
x
x
∈
Σ
p
′
=
0.
Letχ
[
τ
,
s
] be the characteristic function of [
τ
,
s
)
⊆
[
0
,
T
). We can choose χ
[
τ
,
s
]
φ
m
k
m
(
v
-
u
) as the test function, then (35)
∫
τ
s
∫
Ω
φ
m
k
m
v
-
u
∂
v
-
u
∂
t
d
x
d
t
+
∑
i
=
1
n
∫
τ
s
∫
Ω
b
i
x
v
x
i
q
i
x
-
2
v
x
i
-
u
x
i
q
i
x
-
2
u
x
i
v
x
i
-
u
x
i
k
m
′
v
-
u
φ
m
x
d
x
d
t
+
∑
i
=
1
n
∫
τ
s
∫
Ω
b
i
x
v
x
i
q
i
x
-
2
v
x
i
-
u
x
i
q
i
x
-
2
u
x
i
v
x
i
-
u
x
i
k
m
v
-
u
φ
m
x
i
d
x
d
t
=
0
.
Certainly, we have(36)
∫
Ω
b
i
x
v
x
i
q
i
x
-
2
v
x
i
-
u
x
i
q
i
x
-
2
u
x
i
v
x
i
-
u
x
i
k
m
′
v
-
u
φ
m
x
d
x
≥
0
.
Sincev
t
∈
L
2
(
Q
T
), by the Lebesgue dominated convergence theorem, we have (37)
l
i
m
m
→
∞
∫
τ
s
∫
Ω
φ
m
x
k
m
v
-
u
∂
v
-
u
∂
t
d
x
d
t
=
∫
Ω
v
-
u
x
,
s
d
x
-
∫
Ω
v
-
u
x
,
τ
d
x
.
For the last term on the left hand side of (35), obviously, φ
m
x
i
=
m
φ
x
i when x
∈
Ω
∖
Ω
m; in the other places, it vanishes. By condition (31), we have (38)
∫
Ω
b
i
x
v
x
i
q
i
x
-
2
v
x
i
-
u
x
i
q
i
x
-
2
u
x
i
φ
m
x
i
k
m
v
-
u
d
x
=
∫
Ω
∖
Ω
m
b
i
x
v
x
i
q
i
x
-
2
v
x
i
-
u
x
i
q
i
x
-
2
u
x
i
φ
m
x
i
k
m
v
-
u
d
x
≤
c
m
∫
Ω
∖
Ω
m
b
i
x
v
x
i
q
i
x
+
u
x
i
q
i
x
d
x
1
/
p
i
+
∫
Ω
∖
Ω
m
b
i
x
φ
x
i
q
i
x
d
x
1
/
q
i
+
≤
c
∫
Ω
∖
Ω
m
b
i
x
v
x
i
q
i
x
d
x
1
/
p
i
+
+
∫
Ω
∖
Ω
m
b
i
x
u
x
i
q
i
x
d
x
1
/
p
i
+
·
m
∫
Ω
∖
Ω
m
b
i
x
φ
x
i
q
i
x
d
x
1
/
q
i
+
≤
c
∫
Ω
∖
Ω
m
b
i
x
v
x
i
q
i
x
d
x
1
/
p
i
+
+
c
∫
Ω
∖
Ω
m
b
i
x
u
x
i
q
i
x
d
x
1
/
p
i
+
.
Herep
i
(
x
)
=
q
i
(
x
)
/
(
q
i
(
x
)
-
1
) and p
i
+
=
m
a
x
x
∈
Ω
¯
p
i
(
x
).
Then(39)
l
i
m
m
→
∞
∫
τ
s
∫
Ω
b
i
x
v
x
i
q
i
x
-
2
v
x
i
-
u
x
i
q
i
x
-
2
u
x
i
φ
m
x
i
k
m
v
-
u
d
x
d
t
≤
c
l
i
m
m
→
∞
∫
Ω
∖
Ω
m
b
i
x
v
x
i
q
i
x
d
x
1
/
p
i
+
+
∫
Ω
∖
Ω
m
b
i
x
u
x
i
q
i
x
d
x
1
/
p
i
+
=
0
.Now, let m
→
∞ in (35). Then (40)
∫
Ω
v
x
,
s
-
u
x
,
s
d
x
⩽
∫
Ω
v
x
,
τ
-
u
x
,
τ
d
x
,and by the arbitrary of τ, we have (33). The theorem is proved.Corollary 7.
Theorem2 is true.Proof.
We only need to choose(41)
φ
x
=
∏
i
=
1
n
b
i
x
;in Theorem 6, the conclusion is clear.
Certainly, there are many choices ofφ. For example, when x is near to the boundary, φ
(
x
)
=
d
Σ
p
′
(
x
)
=
d
i
s
t
(
x
,
Σ
p
′
).Corollary 8.
Instead of the condition (31), if (42)
m
∫
Ω
∖
Ω
m
b
i
x
d
x
1
/
q
i
+
≤
cand φ
(
x
)
=
d
Σ
p
′, then the same conclusion of Theorem 6 is true.
Only if one notices that(43)
φ
x
i
=
d
x
i
≤
∇
d
=
1
, then the corollary follows.


## 3. Proof of Theorem4

Similar to the proof of Lemma 3.2 in [2], we have the following lemma.Lemma 9.
If for any givenj
∈
{
1,2
,
…
,
n
}, (44)
∫
Ω
b
j
-
1
/
q
j
x
-
1
x
d
x
<
∞
,then (45)
∫
Ω
v
x
j
d
x
≤
c
.
One omits the details of the proof here. By this lemma, one can see that ifb
i
(
x
) satisfies (7), (8), and (44), then one can define the trace of v on the boundary ∂
Ω.Proof of Theorem4.
SinceΣ
p
¯
∩
Σ
p
′
¯
≠
∅, (17) is not true generally. But we have added another condition (44) in Theorem 4; by Lemma 9, we still can impose the partial boundary condition (15). Accordingly, we can choose χ
[
τ
,
s
]
φ
m
k
m
(
v
-
u
) as the test function. Thus, similar to the proof of Theorem 6, we can prove Theorem 4.


## 4. Conclusion

An anisotropic parabolic equation is considered in this paper. In our previous work [2], if the diffusion coefficients are degenerate on the boundary in some directions, while in the other directions they are not degenerate, how to give a suitable partial boundary value condition to match the equation had been studied. In this short paper, we consider the problem in a different view. We assume that the all diffusion coefficients are degenerate on a part of the boundary Σ
p
′ but not degenerate on the remained part of the boundary Σ
p. It is clear that we should impose the boundary value condition on Σ
p. By choosing a test function associated with the domain, the stability of the weak solutions is proved in this paper based on the partial boundary value condition. The method of choosing a test function associated with the domain is an innovative method, which can be generalized to use in the other kinds of the degenerate parabolic equation.



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*Source: 6836417-2018-04-08.xml*