Differential and Integral Equations

Global behaviour of solutions to a parabolic mean curvature equation

Bernd Kawohl and Nickolai Kutev

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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

Permanent link to this document
https://projecteuclid.org/euclid.die/1369056133

Mathematical Reviews number (MathSciNet)
MR1348958

Zentralblatt MATH identifier
0844.35050

Subjects
Primary: 35K60: Nonlinear initial value problems for linear parabolic equations
Secondary: 35B40: Asymptotic behavior of solutions 35D99: None of the above, but in this section 53A10: Minimal surfaces, surfaces with prescribed mean curvature [See also 49Q05, 49Q10, 53C42]

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


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