Self-similar lifting and persistent touch-down points in the thin-film equation

TL;DR

This paper investigates self-similar solutions to the thin-film equation with mobility exponent m in (0,4], revealing mechanisms for lifting touch-down points and non-uniqueness of solutions in certain cases.

Contribution

It introduces a new approach to construct self-similar solutions using a four-dimensional dynamical system and a shooting method, highlighting non-uniqueness for m in (2,4).

Findings

Existence of self-similar solutions for the thin-film equation with specific initial profiles.

Identification of non-uniqueness of solutions with persistent touch-down points for m in (2,4).

Development of a shooting argument based on invariant regions and energy formulas.

Abstract

We study self-similar solutions of the thin-film equation, with mobility exponent m in (0,4], that describe the lifting of an isolated touch-down point given by an initial profile of the form |x|. This provides a mechanism for non-uniqueness of the thin-film equation with m in (2,4), since solutions with a persistent touch-down point also exist in this case. In order to prove existence of the self-similar solutions, we need to study a four-dimensional continuous dynamical system. The proof consists of a shooting argument based on the identification of invariant regions and on suitable energy formulas.