The weak solutions of a nonlinear parabolic equation from two-phase problem

Abstract

Abstract A nonlinear parabolic equation from a two-phase problem is considered in this paper. The existence of weak solutions is proved by the standard parabolically regularized method. Different from the related papers, one of diffusion coefficients in the equation, b ( x ) $b(x)$ , is degenerate on the boundary. Then the Dirichlet boundary value condition may be overdetermined. In order to study the stability of weak solution, how to find a suitable partial boundary value condition is the foremost work. Once such a partial boundary value condition is found, the stability of weak solutions will naturally follow.

Keywords

• Nonlinear parabolic equation
• Two-phase problem
• Partial boundary value condition
• Stability