On the stability of a non-Newtonian polytropic filtration equation

Abstract

Abstract The non-Newtonian polytropic filtration equation with a convection term vt=div(a(x)|v|α|∇v|p−2∇v)+∑i=1N∂ai(v,x,t)∂xi $$ v_{t}= \operatorname{div} \bigl(a(x) \vert v \vert ^{\alpha }{ \vert {\nabla v} \vert ^{p-2}}\nabla v \bigr)+ \sum_{i=1}^{N}\frac{\partial a_{i}(v,x,t)}{\partial x_{i}} $$ is considered, where p>1 $p>1$, α>0 $\alpha >0$, a(x)≥0 $a(x)\geq 0$ with a(x)|x∈∂Ω=0 $a(x) | _{x\in \partial \varOmega }=0$. This kind of equation is degenerate on the boundary, the usual boundary value condition may be overdetermined. Some conditions depending on a(x) $a(x)$ and ai(⋅,x,t) $a_{i}(\cdot ,x,t)$, which can take place of the boundary value condition, are found. Moreover, how the nonlinear term |v|α $|v|^{\alpha }$ affects the stability of weak solutions is revealed.

Keywords

• Non-Newtonian polytropic filtration equation
• Convection term
• Boundary value condition
• Stability