On the Weak Characteristic Function Method for a Degenerate Parabolic Equation

For a nonlinear degenerate parabolic equation, how to impose a suitable boundary value condition to ensure the well-posedness of weak solutions is a very important problem. It is well known that the classical Fichera-Oleinik theory has perfectly solved the problem for the linear case, and the optimal boundary value condition matching up with a linear degenerate parabolic equation can be depicted out by Fechira function. In this paper, a new method, which is called the weak characteristic function method, is introduced. By this new method, the partial boundary condition matching up with a nonlinear degenerate parabolic equation can be depicted out by an inequality from the diffusion function, the convection function, and the geometry of the boundary ∂Ω itself. Though, by choosing different weak characteristic function, one may obtain the differential partial boundary value conditions, an optimal partial boundary value condition can be prophetic. Moreover, the new method works well in any kind of the degenerate parabolic equations.