Relaxation to equilibrium in the one-dimensional thin-film equation with partial wetting

TL;DR

This paper studies the long-term behavior of solutions to a one-dimensional thin-film equation with partial wetting, demonstrating stability of steady states using energy-dissipation relations in Lagrangian coordinates.

Contribution

It introduces a novel analysis of finite-mass solutions for the thin-film equation with partial wetting, establishing stability results not previously shown.

Findings

Steady state solutions are stable over time.

The method uses differential relations between energy and dissipation.

Solutions with finite mass are considered, differing from earlier work.

Abstract

We investigate the large time behavior of compactly supported solutions for a one-dimensional thin-film equation with linear mobility in the regime of partial wetting. We show the stability of steady state solutions. The proof uses the Lagrangian coordinates. Our method is to establish and exploit differential relations between the energy and the dissipation as well as some interpolation inequalities. Our result is different from earlier results because here we consider solutions with finite mass.