Abstract
Consider the anisotropic parabolic equation with the variable exponents where , , and . If is not degenerate on , a part of the boundary, but is degenerate on the remained part , then the boundary value condition is imposed on , but there is no boundary value condition required on . The stability of the weak solutions can be proved based on the partial boundary value condition .
1. Introduction
Recently, we had considered the anisotropic parabolic equation with the initial-boundary value conditions where , , , and is a bounded domain with a smooth boundary . If , the existence of the weak solution had been proved by Antontsev and Shmarev [1]. If some of are degenerate on the boundary, Zhan [2] had conjectured that, instead of the usual boundary value condition (3), only a partial boundary value condition, should be imposed, while is relatively open in . For example, if and has some special restrictions, then the explicit 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 , how to depict out the explicit seems very difficult. In this short paper, we give an original attempt, we first assume that where is a suitable positive constant, and denote that A simple example satisfies (6) is
Definition 1. If a function satisfies that for any function and , then we say 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 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. If satisfies (6), satisfies (7)-(8), and for large enough , and are two solutions of (1) with the same partial boundary value condition then where .
Remark 3. Since the domain satisfies (6) and satisfies (7)-(8) and when is near to , (1) is not degenerate, by (14), then we can define the trace of on , and condition (15) is reasonable. Here is a small enough constant, However, when , then (17) is not clear. In this case, only if then we have a similar conclusion. This is the following theorem.
Theorem 4. If the domain satisfies satisfies (19) and condition (14) is true when is large enough, and if and 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 domain and in Theorem 4. For example, ,
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: which originally comes from the electrorheological fluids theory (see [3, 4]). If , there are many related papers; one can see [5–7] and the references therein. If when but , 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 satisfies some other restrictions.
2. The Stability
The concepts of the exponent variable spaces, , , , and , can be found in [11–15].
Lemma 5 (see [11–13]). If and are real functions with and , then, for any and , one has Moreover, One lets be an odd function, and Then, Let be a function satisfying
Theorem 6. If satisfies (6), satisfies (7)-(8), and for large enough , and are two solutions of (1) with the same partial boundary value conditionthen
Proof. We let and be two weak solutions of (1) with the partial boundary value condition (32). Then, .
LetThen .
Let be the characteristic function of . We can choose as the test function, then Certainly, we have Since , by the Lebesgue dominated convergence theorem, we have For the last term on the left hand side of (35), obviously, when ; in the other places, it vanishes. By condition (31), we have Here and .
Then Now, let in (35). Then and by the arbitrary of , we have (33). The theorem is proved.
Corollary 7. Theorem 2 is true.
Proof. We only need to choose in Theorem 6, the conclusion is clear.
Certainly, there are many choices of . For example, when is near to the boundary, .
Corollary 8. Instead of the condition (31), if and , then the same conclusion of Theorem 6 is true.
Only if one notices that then the corollary follows.
3. Proof of Theorem 4
Similar to the proof of Lemma 3.2 in [2], we have the following lemma.
Lemma 9. If for any given , then One omits the details of the proof here. By this lemma, one can see that if satisfies (7), (8), and (44), then one can define the trace of on the boundary .
Proof of Theorem 4. Since , (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 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 but not degenerate on the remained part of the boundary . It is clear that we should impose the boundary value condition on . 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.
Conflicts of Interest
The author declares that he has no conflicts of interest.
Acknowledgments
The paper is supported by NSF of Fujian Province, China.