On an Anisotropic Parabolic Equation on the Domain with a Disjoint Boundary

Copyright © 2018 Huashui Zhan.This is an open access article distributed under the Creative CommonsAttribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Consider the anisotropic parabolic equation with the variable exponents Vt = ∑ni=1(bi(x)|Vxi |qi(x)−2Vxi )xi ,where bi(x), qi(x) ∈ C1(Ω), qi(x) > 1, and bi(x) ≥ 0. If {bi(x)} is not degenerate on Σp ⊂ ∂Ω, a part of the boundary, but is degenerate on the remained part ∂Ω \ Σp, then the boundary value condition is imposed on Σp, but there is no boundary value condition required on ∂Ω \ Σp. The stability of the weak solutions can be proved based on the partial boundary value condition V|x∈Σp = 0.

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 Ω satisfies (6),   satisfies ( 7)- (8) ) V(, ) and (, ) are two solutions of (1) with the same partial boundary value condition where Remark 3. Since the domain Ω satisfies ( 6) and   satisfies ( 7)-( 8) and when  is near to Σ  , (1) is not degenerate, by (14), then we can define the trace of V(, ) on Σ  , and condition (15) is reasonable.Here  > 0 is a small enough constant, However, when Σ  ∩ Σ   ̸ = 0, 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 Ω satisfies
satisfies ( 19) and condition ( 14) is true when  is large enough, and if V(, ) 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 Ω and   () in Theorem 4. For example,  = 2, 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 () ≡ 1, there are many related papers; one can see [5][6][7] and the references therein.If () > 0 when  ∈ Ω but ()| ∈Ω = 0, then the stability of the weak solutions without the boundary value condition had been studied by Zhan et al. [8][9][10], provided that the diffusion coefficient () satisfies some other restrictions.
Let  [,] be the characteristic function of [, ) ⊆ [0, ).We can choose  [,]     (V − ) as the test function, then Certainly, we have (37) For the last term on the left hand side of (35), obviously,    =    when  ∈ Ω\Ω  ; in the other places, it vanishes.By condition (31), we have then the corollary follows.

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  ∈ {1, 2, . . ., }, 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 V on the boundary Ω.
Proof of Theorem 4. Since Σ  ∩ Σ   ̸ = 0, (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  [,]     (V − ) as the test function.Thus, similar to the proof of Theorem 6, we can prove Theorem 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 Σ   but not degenerate on the remained part of the boundary Σ  .It is clear that we should impose the boundary value condition on Σ  .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.