The Evolutionary p ( x )-Laplacian Equation with a Partial Boundary Value Condition

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 α < p− − 1, 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 α > p+ − 1, the stability of the solutions is obtained without the boundary value condition. At the same time, only if α > 0 and p− > 1 can the uniqueness of the solutions be proved without any boundary value condition.

for any function  ∈  ∞ (0, ;  (Ω)), then we say that (, ) is the weak solution of (6).The initial value (2) is true in the sense of The partial boundary condition (7) is true in the sense of trace.If we 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 0 <  <  − − 1, we can prove that the weak solution  ∈  ∞ (0, ;  1, (Ω)) for some  > 1.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 V(, ) be two weak solutions of (6) with the different initial values  0 () and V 0 (), respectively, and with the same partial homogeneous boundary value If () is a Lipschitz function and then where Σ  is the part of the boundary expressed as (8).
Secondly, if  ≥  − − 1, 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  >  + − 1.
Theorem 3. Let  and V be two weak solutions of (6) with the different initial values (, 0) and V(, 0), respectively.If  >  + − 1, the constant  ≥ max{/ − , 1}, and then for any  ∈ [0, ), there holds Last but not least, no matter whether  <  − − 1 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,  > 0,  − > 1, 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).

The Stability of Solutions When 𝛼<𝑝 − −1
Lemma 5 (see [9]).(i) Let () and () be real functions with 1/() + 1/() = 1 and () > 1.Then, for any  ∈  () (Ω) and V ∈  () (Ω), we have Proof of Theorem 2. For a small positive constant  > 0, let where Then For small  > 0, let Obviously, ℎ  () ∈ (R) and If  and V are two weak solutions of ( 6) with the same partial homogeneous boundary value (13) and   (( − V)) is chosen to be the test function, then Thus Obviously, we have Using the Young inequality, we have which goes to 0 as  → 0, due to the assumption that According to the definition of the trace, by the partial boundary value condition (7), we have lim Moreover, as in [17], we can prove that lim The details of the proof of (33) are omitted here.

The Stability of Solutions without the Boundary Value Condition
As we have said in the introduction, when  ≥  − − 1, 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  satisfies and for any function We first introduced this kind of the weak solutions in our previous paper [19], in which the following equation was studied: where 0 ≤ () ∈  1 (Ω) with ()| ∈Ω = 0.It is not difficult to prove the existence of the weak solution in the sense of Definition 6.