Diffusion Convection Equation with Variable Nonlinearities

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.

Throughout the paper, we assume that 1 < () ∈  1 (Ω) and denote The main aim of our paper is to study the stability based on the partial boundary value condition (4).
If  = 0 in Theorem 1, without condition (7), conclusion (8) is true.In other words, we have the following important result.
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.

The Definition of the Weak Solutions
Here, the basic function spaces with variable exponents are quoted; for more details, see [13][14][15][16] et al.Set For any ℎ ∈  + (Ω) we define For any  ∈  + (Ω), we introduce the variable exponent Lebesgue spaces and the variable exponent Sobolev space.

Lemma 4. (i) The space (𝐿
. 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, then we have and the embedding is compact if inf ∈Ω ( * () − ()) > 0.
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 ,  ∈  + (Ω) 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.
it is not difficult to prove there exists a weak solution in the sense of Definition 7.
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.

The Proofs of Theorems
Proof of Theorem 1.Let , V be two solutions of (1) with the partial homogeneous boundary values (4) and with the initial values  0 , V 0 , respectively.From the definition of the weak solution in the sense of Definition 7, for all  ∈  1 0 (  ), For small  > 0, let (40) By a process of limit, we can choose     ( − V) as the test function; then Thus, by (ii) and (iii) in Lemma 4, which goes to 0 as  → 0 by Here, we had used the fact         1   1/() ∇          () (Ω\Ω  ) by (6).
For any given  ∈ (0, ), let which goes to zero when  → 0 by the assumption that At the same time, Now, let  → 0 in (41).Then It implies that Proof of Theorem 2. From Definition 7, if  = 0, one can see that condition ( 7) is naturally true.In other words, condition (7) is not necessary.

At last
and since   is a Lipschitz function, , V ∈  ∞ (  ), we have where  < 1.By (67), it is easy to obtain the local stability ( 12), and we omit the details here.

Conclusions
The equation considered in the paper comes from electrorheological fluids, which may be double degenerate or singular.Moreover, the diffusion coefficient is degenerate on the boundary; then the solutions generally lack the regularity to define the trace on the boundary.The facts make it difficult to obtain the stability of the weak solutions.By introducing a new kind of the weak solution, the paper successfully overcomes the difficulty.Moreover, importantly, the main result (Theorem 1) shows that the electrorheological fluid theory must be complicated compared to the non-Newtonian fluid theory.