Differential and Integral Equations

Global behaviour of solutions to a parabolic mean curvature equation

Abstract

Consider (1.1) for a domain $\Omega$ for which there is no classical nonparametric solution of the stationary problem. We study viscosity solutions of (1.1). In general they fail to satisfy Dirichlet data on the boundary and "detach." In fact the solution tends to infinity with finite speed. The velocity stabilizes as $t\to \infty$, and we give some results on asymptotic growth. These new effects can be reconciled with the notion of viscosity solutions. The free boundary data are shown to be Lipschitzian for special domains $\Omega$. Problem (1.1) is related to some isoperimetric geometric problem.

Article information

Source
Differential Integral Equations, Volume 8, Number 8 (1995), 1923-1946.

Dates
First available in Project Euclid: 20 May 2013

https://projecteuclid.org/euclid.die/1369056133

Mathematical Reviews number (MathSciNet)
MR1348958

Zentralblatt MATH identifier
0844.35050

Citation

Kawohl, Bernd; Kutev, Nickolai. Global behaviour of solutions to a parabolic mean curvature equation. Differential Integral Equations 8 (1995), no. 8, 1923--1946. https://projecteuclid.org/euclid.die/1369056133