The well-posedness problem of an anisotropic porous medium equation with a convection term

Abstract

The initial boundary value problem of an anisotropic porous medium equation is considered in this paper. The existence of a weak solution is proved by the monotone convergent method. By showing that nabla uin L^{infty}(0,T; L^{2}_{mathrm{loc}}(Omega )), according to different boundary value conditions, some stability theorems of weak solutions are obtained. The unusual thing is that the partial boundary value condition is based on a submanifold Σ of partial Omega times (0,T) and, in some special cases, Sigma = {(x,t)in partial Omega times (0,T): prod a_{i}(x,t)>0 }.