Global exponential stability of stationary profiles in a thin film equation with second-order diffusion

TL;DR

This paper proves that solutions to a thin-film equation with a confinement potential and second-order diffusion converge exponentially to a stationary profile, provided the additional forces are sufficiently weak, using Wasserstein gradient flow methods.

Contribution

It demonstrates the persistence of exponential convergence in a thin-film equation with second-order effects under small force conditions, extending previous results.

Findings

Solutions converge exponentially to stationary profiles

Exponential rate persists under small force conditions

Wasserstein gradient flow framework is effective for analysis

Abstract

We study existence and long-time behavior of weak solutions to a thin-film equation with a confinement potential and a second-order degenerate diffusion term. It is known that in absence of second order effects, solutions for general initial data converge at an exponential rate in time to the unique stationary profile. Our main result is that if the strength of the additional forces is sufficiently small, this global exponential equilibration behavior persists, at a slightly smaller rate. Our proof uses the formulation of the equation as a Wasserstein gradient flow, and an auxiliary lower-order Lyapunov functional.