On a doubly degenerate parabolic equation with a nonlinear damping term

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