The paper follows with interest in a nonlinear parabolic equation coming from the electrorheological fluid ut = div(a(x)|∇u|p(x)−2∇u) + XN i=1 ∂bi(u, x, t) ∂xi with a(x) being positive in Ω. We study the well-posedness problem of the equation under the condition bi(·, x, t) = 0 on the partial boundary ∂Ω Σ1 for every i = 1, 2, · · · ,N, where Σ1 = {x ∈ ∂Ω : 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) ∈ Σ1 × (0, T).