Advances in Differential Equations

Asymptotically self-similar solutions to curvature flow equations with prescribed contact angle and their applications to groove profiles due to evaporation-condensation

Nao Hamamuki

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Abstract

We study the asymptotic behavior of solutions to fully nonlinear second order parabolic equations including a generalized curvature flow equation which was introduced by Mullins in 1957 as a model of evaporation-condensation. We prove that, in the multi-dimensional half space, solutions of the problem with prescribed contact angle asymptotically converge to a self-similar solution of the associated problem under a suitable rescaling. Several properties of the profile function of the self-similar solution are also investigated. We show that the profile function has a corner and that the angles are determined by points at which the equation is degenerate. We also study the depth of the groove, which is represented by the value of the profile function at the boundary. Among other results it turns out that, as the contact angle tends to zero, the depth of the groove is well approximated by the linearized problem.

Article information

Source
Adv. Differential Equations Volume 19, Number 3/4 (2014), 317-358.

Dates
First available in Project Euclid: 30 January 2014

Permanent link to this document
https://projecteuclid.org/euclid.ade/1391109088

Mathematical Reviews number (MathSciNet)
MR3161664

Zentralblatt MATH identifier
1295.35089

Subjects
Primary: 35B40: Asymptotic behavior of solutions 35C06: Self-similar solutions 35D40: Viscosity solutions 35K20: Initial-boundary value problems for second-order parabolic equations

Citation

Hamamuki, Nao. Asymptotically self-similar solutions to curvature flow equations with prescribed contact angle and their applications to groove profiles due to evaporation-condensation. Adv. Differential Equations 19 (2014), no. 3/4, 317--358. https://projecteuclid.org/euclid.ade/1391109088.


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