Strong solutions of the thin film equation in spherical geometry

TL;DR

This paper investigates the existence and long-term behavior of strong solutions to a spherical thin film equation, demonstrating that solutions tend to flatten over time, using advanced mathematical estimates.

Contribution

It provides new existence results and decay properties for solutions to the doubly degenerate fourth-order parabolic equation on a spherical surface.

Findings

Strong solutions exist under certain conditions.

Solutions asymptotically decay to a flat profile.

The analysis employs weighted Sobolev space estimates.

Abstract

We study existence and long-time behaviour of strong solutions for the thin film equation using a priori estimates in a weighted Sobolev space. This equation can be classified as a doubly degenerate fourth-order parabolic and it models coating flow on the outer surface of a sphere. It is shown that the strong solution asymptotically decays to the flat profile.