On the partial boundary value condition basing on the diffusion coefficient

Abstract

\ The paper follows with interest in a nonlinear parabolic equation coming from the electrorheological fluid

{u_t}= \operatorname{div} (a(x){\left| {\nabla u} \right|^{p(x) - 2}}\nabla u)+ \sum\limits_{i = 1}^N {\frac{{\partial {b_i}(u,x,t)}}{{\partial {x_i}}}}

with $a(x)$ being positive in $\Omega$. We study the well-posedness problem of the equation under the condition $b_i(\cdot, x,t)=0$ on the partial boundary $\partial \Omega\setminus\Sigma_1$ for every $i=1,2,\cdots, N$, where $\Sigma_1=\{x\in \partial \Omega: a(x)>0\}$. The stability of the weak solutions is obtained only basing on a partial boundary value condition $u(x,t)=0, (x,t)\in \Sigma_1\times (0,T)$.