Asymptotically self-similar graph-like solutions to a multi-dimensional surface diffusion flow equation under contact angle and no-flux boundary conditions

TL;DR

This paper proves the existence and convergence of self-similar solutions to a multi-dimensional surface diffusion flow model with contact angle and no-flux boundary conditions, under near-critical initial slopes.

Contribution

It establishes the existence of unique global solutions and their asymptotic self-similarity without restrictions on the contact angle size.

Findings

Existence of a unique global-in-time solution.

Convergence to a self-similar solution over time.

No size restriction on the contact angle.

Abstract

This paper studies Mullins' model of thermal grooving which consists of a surface diffusion flow equation with contact angle and no-flux boundary conditions. We consider this problem in a multi-dimensional half space and prove that if the slope of the initial data is close to that consistent with the contact angle, then there exists a unique global-in-time solution. In particular, we show the existence of a self-similar solution for a given behavior at the space infinity. We also show that our global solution converges to a self-similar solution as the time tends to infinity if the initial data is asymptotically homogeneous at the space infinity. No assumption on the size of the contact angle is imposed.