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
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
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.