Affiliation:
1. School of Applied Mathematics, Xiamen University of Technology, Xiamen 361024, China
2. School of Science, Jimei University, Xiamen 361021, China
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 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.
Funder
Natural Science Foundation of Fujian Province