摘要：Consider a double degenerate parabolic equation arising from the electrorheological fluids theory and many other diffusion problems. Let $v_{\varepsilon }$ be the viscous solution of the equation. By showing that $ \nabla v_{\varepsilon } \in L^{\infty }(0,T; L_{\mathrm{loc}}^{p(x)}(\Omega ))$ and $\nabla v_{\varepsilon }\rightarrow \nabla v$ almost everywhere, the existence of weak solutions is proved by the viscous solution method. By imposing some restriction on the nonlinear damping terms, the stability of weak solutions is established. The innovation lies in that the homogeneous boundary value condition is substituted by the condition $a(x) _{x\in \partial \Omega }=0$ , where $a(x)$ is the diffusion coefficient. The difficulties come from the nonlinearity of $\vert {\nabla v} \vert ^{p(x)-2}$ as well as the nonlinearity of $ v ^{\alpha (x)}$ .
