open access

https://doi.org/10.14943/112040

https://hdl.handle.net/2115/93633

LARGE TIME BEHAVIOR OF EXPONENTIAL SURFACE DIFFUSION FLOWS ON R

英語

2024-11-28

Department of Mathematics, Hokkaido University

Hokkaido University Preprint Series in Mathematics

1161

1-23

We consider a surface diffusion flow of the form V = ∂2s f (−κ) with a strictly increasing smooth function f typically, f (r) = er , for a curve with arc-length parameter s, where κ denotes the curvature and V denotes the normal velocity. The conventional surface diffusion flow corresponds to the case when f (r) = r. We consider this equation for the graph of a function defined on the whole real line R. We prove that there exists a unique global-in-time classical solution provided that the first and the second derivatives are bounded and small. We further prove that the solution behaves like a solution to a self-similar solution to the equation V = −f ′(0)κ. Our result justifies the explanation for grooving modeled by Mullins (1957) directly obtained by Gibbs-Thomson law without linearization of f near κ = 0.

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