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 nonlin...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Hindawi Limited
2018-01-01
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| Series: | Discrete Dynamics in Nature and Society |
| Online Access: | http://dx.doi.org/10.1155/2018/1237289 |
| Summary: | 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. |
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| ISSN: | 1026-0226 1607-887X |