The Evolutionary -Laplacian Equation with a Partial Boundary Value Condition

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 equationcomes 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 valueand the homogeneous boundary valuehas 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 equationin [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 conditionwhere , , , 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 ofThe partial boundary condition (7) is true in the sense of trace.
Ifwe 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 valueIf is a Lipschitz function andthenwhere 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 , letwhereThenFor small , letObviously, andIf and are two weak solutions of (6) with the same partial homogeneous boundary value (13) and is chosen to be the test function, thenThusObviously, we haveUsing the Young inequality, we havewhich goes to 0 as , due to the assumption that implies thatwhile Since is a Lipschitz function, . According to the definition of the trace, by the partial boundary value condition (7),we haveMoreover, as in [17], we can prove thatThe 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 haveand by the Gronwall inequality, we haveTheorem 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 satisfiesand for any function ,   such that for any given ,  ,and the initial value (2) is satisfied in the sense ofWe 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, sinceby leading to , (43) yieldsIn the second place, since , by (16), , thenBy and the condition (16), we haveby 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 havewhere .
Once more,Let in (42). By (44), (45), (49), and (50), we haveLet . Without loss of the generality, we may assume that there exist and . Then for any , . If we denote thatthen andBy , there exists a constant such thatBy (51) and (54), we haveand using the Gronwall inequality, we havewhere depends on . Thus, we have The proof is complete.

4. The Partial Boundary Value Condition

Consider the linear degenerate equationwhich 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, iswhere 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.