Abstract

The paper studies diffusion convection equation with variable nonlinearities and degeneracy on the boundary. Unlike the usual Dirichlet boundary value, only a partial boundary value condition is imposed. If there are some restrictions in the diffusion coefficient, the stability of the weak solution based on the partial boundary value condition is obtained. In general, we may obtain a local stability of the weak solutions without any boundary value condition.

1. Introduction

Consider the following diffusion convection equation with variable nonlinearities: where is a bounded domain with smooth boundary and is a measurable function. Equation (1) comes from the so-called electrorheological fluids, comes from a motion of an ideal barotropic gas through a porous medium, and comes from the flows in fractured media, and so on (see [1, 2]). If , one can impose the following initial-boundary value conditions: and there are many references recently. We would like to suggest that the first paper where parabolic equations with variable growth exponent are considered is Acerbi et al. [3]; some other interesting works are listed as [4–11] in our references. If , then the equation is degenerate on the boundary; the author had shown that, in [12], besides the initial value condition (2), only a partial boundary value conditionis imposed, where is a relatively open subset. In some cases, ; then the solutions are determined by the initial value completely.

Throughout the paper, we assume that and denote The main aim of our paper is to study the stability based on the partial boundary value condition (4).

Theorem 1. Suppose that , when , and when . Let be two solutions of (1) with the same partial boundary value condition (4) and with the different initial values , , respectively. If for small positive , and then Here, .

If , , the two restrictions of in (6) are incompatible, and implies , while implies that . So, if , it is impossible to obtain Theorem 1 (also Theorem 2). However, is a continuous function; when , the two restrictions of in (6) are compatible; for example, if , by (6), satisfies One can see that only if , is large enough, and (9) is true.

If in Theorem 1, without condition (7), conclusion (8) is true. In other words, we have the following important result.

Theorem 2. Suppose that when and when . Let be two solutions of equation with the same partial boundary value condition (4) and with the different initial values , , respectively. Then stability (8) is true only if satisfies (6).

Theorem 2 (also Theorem 1) has shown an essential difference between (1) and the usual evolutionary Laplacian equation. For the usual evolutionary Laplacian equation, to obtain the stability of the weak solutions, the Dirichlet boundary value condition (3) is necessary.

In general, if does not satisfy conditions (6), we have the following local stability.

Theorem 3. Suppose that , when , and when . Let be two solutions of (1) with the initial values , , respectively. If then there exists a constant such that which implies that (1) with the initial value (3) is unique.

2. The Definition of the Weak Solutions

Here, the basic function spaces with variable exponents are quoted; for more details, see [13–16] et al. Set For any we define For any , we introduce the variable exponent Lebesgue spaces and the variable exponent Sobolev space.

(1) Spaceand it is equipped with the following Luxemburg’s norm: The space is a separable, uniformly convex Banach space.

(2) Spaceand it is endowed with the following norm: We use to denote the closure of in .

Some properties of the function spaces are quoted in the following lemma.

Lemma 4. (i) The space ,  , and are reflexive Banach spaces.
(ii)  -Hölder’s inequality: let and be real functions with and . Then, the conjugate space of is . And, for any and , we have (iii)(iv) If , then (v) If , then (vi) -Poincarés inequality: if , then there is a constant , such that This implies that and are equivalent norms of .
In [14], Zhikov showed that Hence, the property of the space is different from the case when is a constant. This fact can make the general methods used in studying the well-posedness of the solutions to the evolutionary Laplacian equation not be used directly.
If the exponent is required to satisfy logarithmic Hölder continuity condition, with then

Lemma 5. Let be an open, bounded set with Lipschitz boundary and with satisfy the log-Hölder continuity (25). If with satisfies then we have and the embedding is compact if .

Remark 6. Furthermore, under the same assumptions as in the above lemma, if we remove the log-Hölder continuity condition (25), then there is also a continuous and compact embedding where and

Definition 7. A function is said to be a weak solution of (1) with the initial value (2) and the partial boundary value condition (4), if and for any function , The initial value (2) is satisfied in the sense of The partial boundary value condition (4) is satisfied in the sense of the trace.
If satisfies it is not difficult to prove there exists a weak solution in the sense of Definition 7.

Definition 8. A function is said to be a solution of (1) with the initial value (2), if satisfies (31) andwhere , and if we denote that then, for any given , and, for any given , . The initial value (2) is satisfied in the sense of (33).

Based on the existence of the weak solution in the sense of Definition 7, one also can be able to prove the existence of the weak solution in the sense of Definition 8. Since we mainly are concerned with the stability of the weak solutions, we are not ready to give the proof of the existence of the weak solutions in what follows.

3. The Proofs of Theorems

Proof of Theorem 1. Let be two solutions of (1) with the partial homogeneous boundary values (4) and with the initial values , , respectively. From the definition of the weak solution in the sense of Definition 7, for all , For small , let Obviously , and Let By a process of limit, we can choose as the test function; then Thus Thus, by (ii) and (iii) in Lemma 4, Here, or according to (iii) of Lemma 4; also has the same meaning if we denote that .
Using the mean theorem, by (7) and , which goes to zero as by the Lebesgue dominated convergence theorem. Then, letting in (43), we have Let . which goes to as by Here, we had used the fact by (6).
For any given , let . Then which goes to zero when by the assumption that At the same time, Now, let in (41). Then It implies that