On the Weak Characteristic Function Method for a Degenerate Parabolic Equation

Huashui Zhan

doi:10.1155/2019/9040284

Abstract

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.

Highlights

For the earliest movement differential equation of a particle dx dt = f (t, x) (1)the initial value x (0) = x0 (2)is the initial position of the particle
For a degenerate parabolic equation, how to impose a suitable partial boundary value condition to ensure the well-posedness of weak solutions has been an interesting and important problem for a long time
The first discovery of this paper is that, by the weak characteristic new method, we find that the partial boundary value condition (20) admits the form as i=1 where the constant γ satisfies

Summary

Introduction

For a degenerate parabolic equation, how to impose a suitable partial boundary value condition to ensure the well-posedness of weak solutions has been an interesting and important problem for a long time. The author of [12] extended the Dirichlet problem for hyperbolic equations to the L∞ setting and proved the uniqueness of the entropy solution by introducing an integral formulation of the boundary condition This idea had been generalized to deal with the strongly degenerate parabolic equations [13,14,15,16], in which the boundary condition is not directly shown as (27) in sense of the trace but is implicitly contained in a family of entropy inequalities. We can not prove this conjecture for the time being

Main Results

The Proof of Theorem 4

The Proof of Theorem 6