@article{Zhan2022, 
author = {Huashui Zhan and Yuan Zhi and Xiaohua Niu},
title = {On a non-Newtonian fluid type equation with variable diffusion coefficient},
year = {2022},
journal = {AIMS Mathematics},
volume = {7},
number = {10},
pages = {17747-17766},
keywords = {the partial boundary value condition, variable diffusion, non-Newtonian fluid equation, submanifold, trace},
url = {https://www.sciopen.com/article/10.3934/math.2022977},
doi = {10.3934/math.2022977},
abstract = {Since the non-Newtonian fluid type equations arise from a broad and in-depth background, many research achievements have been gained from 1980s. Different from the usual non-Newtonian fluid equation, there is a nonnegative variable diffusion in the equations considered in this paper. Such a variable diffusion reflects the characteristic of the medium which may not be homogenous. By giving a generalization of the Gronwall inequality, the stability and the uniqueness of weak solutions to the non-Newtonian fluid equation with variable diffusion are studied. Since the variable diffusion may be degenerate on the boundary    ∂  Ω, it is found that a partial boundary value condition imposed on a submanifold of    ∂  Ω  ×  (  0  ,  T  ) is enough to ensure the well-posedness of weak solutions. The novelty is that the concept of the trace of    u  (  x  ,  t  ) is generalized by a special way.}
}