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 $|v|^{\alpha (x)}$ .

中文翻译：

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具有非线性阻尼项的双退化简抛物方程

考虑由电流变流体理论和许多其他扩散问题引起的双简并抛物线方程。令$ v _ {\ varepsilon} $为方程的粘性解。通过在L ^ {\ infty}（0，T; L _ {\ mathrm {loc}} ^ {p（x）}（\ Omega））$和$ \中显示$ | \ nabla v _ {\ varepsilon} | \ nabla v _ {\ varepsilon} \ rightarrow \ nabla v $几乎到处都有，通过粘性解法证明了弱解的存在。通过对非线性阻尼项施加一些限制，建立了弱解的稳定性。创新之处在于，齐次边值条件由条件$ a（x）|代替。_ {x \ in \ partial \ Omega} = 0 $，其中$ a（x）$是扩散系数。