Discrete Dynamics in Nature and Society
Volume 2018 (2018), Article ID 1237289, 7 pages
https://doi.org/10.1155/2018/1237289

Research Article

The Evolutionary -Laplacian Equation with a Partial Boundary Value Condition

Huashui Zhan¹ and Zhen Zhou²

¹School of Applied Mathematics, Xiamen University of Technology, Xiamen 361024, China
²School of Science, Jimei University, Xiamen 361021, China

Correspondence should be addressed to Huashui Zhan; huashuizhan@163.com

Received 15 January 2018; Accepted 6 March 2018; Published 18 April 2018

Academic Editor: Chris Goodrich

Copyright © 2018 Huashui Zhan and Zhen Zhou. 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.

Abstract.

Consider a diffusion convection equation coming from the electrorheological fluids. If the diffusion coefficient of the equation is degenerate on the boundary, generally, we can only impose a partial boundary value condition to ensure the well-posedness of the solutions. Since the equation is nonlinear, the partial boundary value condition cannot be depicted by Fichera function. In this paper, when , an explicit formula of the partial boundary on which we should impose the boundary value is firstly depicted. The stability of the solutions, dependent on this partial boundary value condition, is obtained. While , the stability of the solutions is obtained without the boundary value condition. At the same time, only if and can the uniqueness of the solutions be proved without any boundary value condition.

1. Introduction and the Main Results

The evolutionary -Laplacian equation

comes from a new interesting kind of fluids, the so-called electrorheological fluids (see [1, 2]), where is a bounded domain with smooth boundary and is a measurable function. Equation (1) with the initial value

and the homogeneous boundary value

has been researched widely; one can refer to [3–6] et al.

If ,

was considered by Yin and Wang [7], where may be degenerate. Instead of the usual boundary condition (3), they classified the boundary into three parts: the nondegenerate boundary, the weakly degenerate boundary, and the strongly degenerate boundary, by means of a reasonable integral description. The boundary value condition should be supplemented definitely on the nondegenerate boundary and the weakly degenerate boundary. Even earlier, they had studied a simpler equation

in [8]. Here , , and . They showed that only if , the usual boundary value condition (3) can be imposed; while , the uniqueness of the solution can be proved without any boundary value condition.

In this paper, we will consider the evolutionary equation

with the initial value (2) and with a partial boundary condition

where , , , and

In the context, is a function and the definitions of the function spaces with variable exponents can be found in [9–16] et al. We denote that

The weak solutions are defined as follows.

Definition 1. If a function satisfies

for any function , then we say that is the weak solution of (6). The initial value (2) is true in the sense of

The partial boundary condition (7) is true in the sense of trace.
If

we add some restrictions on , by a similar method to that in [17, 18] and the existence of the weak solution of (6) with the initial value (2) can be proved. If , we can prove that the weak solution for some . Then the existence of the weak solutions of (6) with the initial-boundary value conditions can be obtained. The main aim of this paper is to study the stability of the weak solutions.
Firstly, we mainly pay close attention to the stability of the weak solutions based on the partial boundary value conditions.

Theorem 2. Let be a Lipschitz function and and be two weak solutions of (6) with the different initial values and , respectively, and with the same partial homogeneous boundary value

If is a Lipschitz function and

then

where is the part of the boundary expressed as (8).

Secondly, if , the weak solutions of (6) cannot be defined as the trace on the boundary and the boundary value condition cannot be used. In order to overcome this difficulty, we will introduce a new kind of the weak solutions in Section 3 and the stability of the weak solutions can be proved when .

Theorem 3. Let and be two weak solutions of (6) with the different initial values and , respectively. If , the constant , and

then for any , there holds

Last but not least, no matter whether or not, the uniqueness of the weak solutions is always true. Actually, similar as [19], we can prove the following theorem.

Theorem 4. If is a Lipschitz function, , then the solution of (6) with the initial value (2) is unique.

However, for the simplicity of the paper, we will not give the details of the proof of Theorem 4 in what follows.

The rest of the paper is arranged as follows. In Section 2, Theorem 2 is proved. In Section 3, Theorem 3 is proved. In the last section, we give an explanation of the partial boundary value condition (8), and some conclusions similar to Theorem 2 are obtained without condition (14).

2. The Stability of Solutions When

Lemma 5 (see [9]). (i) Let and be real functions with and . Then, for any and , we have

(ii)

Proof of Theorem 2. For a small positive constant , let

where

Then

For small , let

Obviously, and

If and are two weak solutions of (6) with the same partial homogeneous boundary value (13) and is chosen to be the test function, then

Thus

Obviously, we have

Using the Young inequality, we have

which goes to 0 as , due to the assumption that implies that

while

Since is a Lipschitz function, . According to the definition of the trace, by the partial boundary value condition (7),

we have

Moreover, as in [17], we can prove that

The details of the proof of (33) are omitted here.
Once again,

Now, after letting , let in (25). Then, by (26), (28), (32), (33), and (34), we have

and by the Gronwall inequality, we have

Theorem 2 is proved.

3. The Stability of Solutions without the Boundary Value Condition

As we have said in the introduction, when , since the weak solutions of (6) generally lack the regularity, we cannot define the trace on the boundary. Thus, we cannot use the boundary value condition to research the stability or the uniqueness of the weak solution. In order to overcome this difficulty, we introduce another kind of the weak solutions as follows.

Definition 6. A function is said to be a weak solution of (6) with the initial value (2), if satisfies

and for any function ,   such that for any given ,  ,

and the initial value (2) is satisfied in the sense of

We first introduced this kind of the weak solutions in our previous paper [19], in which the following equation was studied:

where with . It is not difficult to prove the existence of the weak solution in the sense of Definition 6.

Proof of Theorem 3. For any fixed , after an approximate procedure, we may choose as a test function in equality (38), where is the characteristic function on and is defined as (20). Thus we have

where .
We can rewrite (41) as follows:

In the first place, since

by leading to , (43) yields

In the second place, since , by (16), , then

By and the condition (16), we have

by which . Here or and   or according to (iii) of Lemma 5.
We denote that

and then

where or according to (iii) of Lemma 5.
Combining (48) with (46), we have

where .
Once more,

Let in (42). By (44), (45), (49), and (50), we have

Let . Without loss of the generality, we may assume that there exist and . Then for any , . If we denote that

then and

By , there exists a constant such that

By (51) and (54), we have

and using the Gronwall inequality, we have

where depends on . Thus, we have

The proof is complete.

4. The Partial Boundary Value Condition

Consider the linear degenerate equation

which is a particular case of (6) (where , ). Rewrite it as

According to the Fichera-Oleinik theory [20, 21], besides the initial value condition (2), since , the partial boundary, where we should impose the boundary value condition, is

where is the inner normal vector of .

By reviewing the formula of (8), One can see that

By this token, condition (13) in Theorem 2 is reasonable.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

The paper is supported by NSF of Fujian Province (no. 2015J01592), China.

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