A New Method to Deal with the Stability of the Weak Solutions for a Nonlinear Parabolic Equation

Abstract

Consider the nonlinear parabolic equation / −div( ( )|∇ |−2 ∇ ) = ( , , , ∇ ) with ( )| ∈Ω > 0 and ( ) ∈ Ω = 0. Though it is well known that the degeneracy of ( ) may cause the usual Dirichlet boundary value condition to be overdetermined, and only a partial boundary value condition is needed, since the nonlinearity, this partial boundary can not be depicted out by Fichera function as in the linear case. A new method is introduced in the paper; accordingly, the stability of the weak solutions can be proved independent of the boundary value condition.