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Asymptotically self-similar solutions to curvature flow equations with prescribed contact angle and their applications to groove profiles due to evaporation-condensation. (English) Zbl 1295.35089

The authors study the asymptotic behavior of solutions of a fully nonlinear second order parabolic equations in half space with Neumann boundary condition. This type of equation which includes the generalized curvature flow equation was first introduced by Mullins in 1957 as a model of evaporation-condensation. They prove that solutions of the problem with prescribed contact angle asymptotically converge to a self-similar solution of the associated problem under a suitable scaling. Further, they prove that the profile function has a corner and that the angles are determined by points at which the equation is degenerate. They also prove that as the contact angle decreases to zero the depth of the groove can be approximated by the corresponding linearized problem.

MSC:

35B40 Asymptotic behavior of solutions to PDEs
35C06 Self-similar solutions to PDEs
53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010)
35K20 Initial-boundary value problems for second-order parabolic equations
35K55 Nonlinear parabolic equations
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Full Text: Euclid