Optimal partial boundary condition for degenerate parabolic equations

Abstract

For the stability of the non-Newtonian fluid equation∂u∂t−div(a(x)|∇u|p−2∇u)−∑i=1Nbi(x)Diu+c(x,t)u=f(x,t), where a(x)|x∈Ω>0, a(x)|x∈∂Ω=0 and bi(x)∈C1(Ω‾), we know that the degeneracy of a(x) may make the usual Dirichlet boundary value condition overdetermined and only a partial boundary value condition is expected. How to depict the geometric characteristic of the partial boundary value condition has been a long-time standing open problem. In this study, an optimal partial boundary value condition has been proposed, and the stability of weak solutions based on this partial boundary value condition is established. When the rate of the diffusion coefficient decays to zero, we explore how it affects the stability of weak solutions.