A New Method to Deal with the Stability of the Weak Solutions for a Nonlinear Parabolic Equation

Abstract

Consider the nonlinear parabolic equation with and . Though it is well known that the degeneracy of may cause the usual Dirichlet boundary value condition to be overdetermined, and only a partial boundary value condition is needed, since the nonlinearity, this partial boundary can not be depicted out by Fichera function as in the linear case. A new method is introduced in the paper; accordingly, the stability of the weak solutions can be proved independent of the boundary value condition.

1. Introduction and the Main Results

The nonlinear parabolic equation comes from the theory of non-Newtonian fluid and had been studied widely; one can refer to [1–6] and the references therein. Here , , , and is a bounded domain with a smooth boundary . If , (1) with the initial valueand with a partial boundary value conditionhas been studied in [7–10]. What catches our attention is since is degenerate on the boundary, to obtain the regularity of the weak solutions on the boundary becomes difficult, and the trace on the boundary can not be defined in the classical sense. Accordingly, how to construct a suitable function to obtain the stability of the weak solutions becomes formidable. In what follows, we will overcome the difficulty by a new method, and the new method is called the general characteristic function method.

For simplicity, we only consider a special case of (1),Certainly, the general characteristic method also can be used to study the stability of weak solutions to (1). We assume that , and are functions, and

Definition 1. A function is said to be a weak solution of (4) with the initial value (2), if and for any function , , , ,The initial value is satisfied in the sense of that

The existence of the solution can be proved in a similar way as that of the evolutionary –Laplacian equation [4]; we omit the details here. In order to study the stability of the weak solutions, let us introduce a new concept.

Definition 2. A nonnegative continuous function is said to be a general characteristic function of , if and only if thatOne can see that Definition 2 is inspired by the usual characteristic function of , which is defined as Unlike , the general characteristic function is not unique. For example, the distance function and the diffusion function defined in (5) both are the general characteristic functions. Actually, we only borrow the concept of the characteristic function , but no more than that. The main results of the paper are the following theorems.

Theorem 3. Let and be two weak solutions of (4) with the initial values and , respectively, andIf there exists a general characteristic function such thatthen

Theorem 4. Let and be two nonnegative solutions of (4) with the initial values and , respectively. If ,and there exists a general characteristic function such thatthen the stability of the weak solutions is true in the sense of (13). Here, .

A local stability of the weak solutions is given as follows.

Theorem 5. Let and be two solutions of (4) with the differential initial values and , respectively. If , and there exists a general characteristic function such that then

From my own perspective, the geometric characteristic of the domain and the degeneracy of the diffusion coefficient can take place of the usual boundary value condition The proofs of Theorems 3–5 are based on the general characteristic function, and we call this method as the general characteristic function method.

Moreover, if we choose different general characteristic functions , we can obtain different results. For example, if we choose , corresponding to Theorems 3–5, we have the following results.

Theorem 6. Let and be two weak solutions of (4) with the initial values and , respectively; suppose and (11) is true, andthen the stability (13) is true.

Theorem 7. Let and be two nonnegative solutions of (4) with the initial values and , respectively. If , (14) is true, then the stability of the weak solutions is true in the sense of (13).

This is due to the fact that only we choose , and condition (15) is natural.

Theorem 8. Let and be two solutions of (4) with the differential initial values and , respectively. If , then there exists a constant such that

This is due to the fact that if , condition (16) is natural. When , only we choose , and condition (16) is also true. Once more, since , condition (17) is natural. At the same time, if , we can choose such that condition (18) is true. When , we can choose such that (18) is true. Then we have the conclusion of Theorem 8.

If we choose , corresponding to Theorems 3–5, we have the following corollaries.

Corollary 9. Let and be two solutions of (4) with the differential initial values and , respectively. Suppose and (11) is true, and satisfies then the stability (13) is true.
In particular, if , (23) becomes that Thus, if , (24) is naturally true. Accordingly, only the condition (11) can ensure that the stability (13) is true naturally.

Corollary 10. Let and be two solutions of (4) with the differential initial values and , respectively. If , (14) is true, and then the stability of the weak solutions is true in the sense of (13). Here .

Corollary 11. Let and be two solutions of (4) with the differential initial values and , respectively. If , and and satisfythen

This is due to the fact that only if we choose , conditions (16) and (18) in Theorem 5 are naturally true.

As long as you like, you can choose the other general characteristic functions; for example, and , to obtain the corresponding theorems. Here, . By the way, we hope the method introduced in this paper can be beneficial to the well-posedness problems for the evolutionary –Laplacian equations, overdetermined anisotropic elliptic equations, and the infiltration equations. As we know, there are many papers devoted to these equations; one can refer to [11–23] and the references therein.

2. The Proof of Theorem 3

For small , let

Obviously , and

Proof of Theorem 3. Let and be two solutions of (4) with the different initial values and , respectively. We choose as the test function in Definition 1. Let be a general characteristic function. ThenLet us analyze every term in (30).If has 0 measure, sinceconsequently If has a positive measure, then by the Lebesgue dominated convergence theoremBy (29) and condition (12), using the Lebesgue dominated convergence theorem, in both cases, we haveAt the same time, by (11), ,Then Moreover, by ,Therefore, we haveNow, let in (30). ThenBy the Gronwall inequality, we have Theorem 3 is proved.

3. The Proof of Theorem 4

Proof of Theorem 4. By Definition 1, for any , , , we haveLet be a general characteristic function, as before, and let Now, we choose in (46); accordingly,Clearly,Since (15), then By (50), (52), using the Hölder inequality, we haveThusMoreover, by , using (29) and Lebesgue dominated convergence theorem, The last equality of (55) is based on the following analysis:while, since , , using Lebesgue dominated convergence theorem, we have and by , ,Moreover,by (52) and , while is obviously true.
By (55), (59), and (60),At last,Now, after letting , let in (47). Then, using (49), (54), (55), (59), (60), (61), and (62), and by the Gronwall inequality, we have

4. The Proof of Theorem 5

Proof of Theorem 5. Let and be two solutions of (4) with the initial values and , respectively. From the definition of the weak solution, if , for any , ,In particular, we choose where is the characteristic function on and is a general characteristic function. Denoting , thenClearly, In the first place, for the second term on the right-hand side of (66), we have Now, since , we have If ,If , by Hölder inequality In the second place, by (17)-(18)since , we easily see thatIn the third place, we have At last, by (66), (67), (68), (69), (70), (71), (73), (74), and (75), we let in (64). Then where . By (76), we easily show thatThus, by the arbitrary of , we haveThe proof is complete.

Data Availability

No data were used to support this study.

Conflicts of Interest

The author declares that he has no conflicts of interest.

Acknowledgments

The paper is partially supported by NSF of Fujian Province and partially supported by NSF of Xiamen University of Technology, China.