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A Class of Parabolic Equations Driven by the Mean Curvature Flow

Part of: Partial differential equations on manifolds; differential operators Parabolic equations and systems Global differential geometry Classical differential geometry

Published online by Cambridge University Press:  30 August 2018

Anderson L. A. de Araujo  and
Marcelo Montenegro
Anderson L. A. de Araujo*
Affiliation:
Departamento de Matemática Universidade Federal de Viçosa, CCE, Avenida PH Rolfs, s/n CEP 36570-900 Viçosa, MG, Brazil (anderson.araujo@ufv.br)
Marcelo Montenegro
Affiliation:
Departamento de Matemática, Rua Sérgio Buarque de Holanda, Universidade Estadual de Campinas, IMECC, 651 CEP 13083-859 Campinas, SP, Brazil (msm@ime.unicamp.br)
*
*Corresponding author.
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Abstract

We study a class of parabolic equations which can be viewed as a generalized mean curvature flow acting on cylindrically symmetric surfaces with a Dirichlet condition on the boundary. We prove the existence of a unique solution by means of an approximation scheme. We also develop the theory of asymptotic stability for solutions of general parabolic problems.

Keywords

mean curvature flowasymptotic behaviourstability

MSC classification

Primary: 35K55: Nonlinear parabolic equations
Secondary: 35K57: Reaction-diffusion equations 53C44: Geometric evolution equations (mean curvature flow, Ricci flow, etc.) 53A04: Curves in Euclidean space 53A05: Surfaces in Euclidean space 58J35: Heat and other parabolic equation methods
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 62 , Issue 1 , February 2019 , pp. 135 - 163
Copyright
Copyright © Edinburgh Mathematical Society 2018 

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