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Discrete Dynamics in Nature and Society
Volume 2018, Article ID 1237289, 7 pages
https://doi.org/10.1155/2018/1237289
Research Article

The Evolutionary -Laplacian Equation with a Partial Boundary Value Condition

1School of Applied Mathematics, Xiamen University of Technology, Xiamen 361024, China
2School of Science, Jimei University, Xiamen 361021, China

Correspondence should be addressed to Huashui Zhan; moc.361@nahziuhsauh

Received 15 January 2018; Accepted 6 March 2018; Published 18 April 2018

Academic Editor: Chris Goodrich

Copyright © 2018 Huashui Zhan and Zhen Zhou. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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 , 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 , the stability of the solutions is obtained without the boundary value condition. At the same time, only if and can the uniqueness of the solutions be proved without any boundary value condition.