On self-similar collapse of discontinuous data for thin film equations with doubly degenerate mobility

TL;DR

This paper demonstrates the existence of self-similar solutions for a thin film equation with discontinuous initial data, considering both free boundary and Cauchy problem settings, highlighting the equation's degeneracy at specific points.

Contribution

It establishes the existence of self-similar solutions for a degenerate thin film equation with discontinuous data, covering both FBP and Cauchy problem scenarios.

Findings

Self-similar solutions exist for the degenerate thin film equation with discontinuous data.

Both free boundary and Cauchy problem frameworks are addressed.

Solutions exhibit specific behaviors near interfaces, including oscillations.

Abstract

It is shown that a Riemann-type problem with discontinuous data of sign-type for the thin film equation, which degenerates at +1 and -1, admits a self-similar solution. Both FBP and the Cauchy problem (oscillatory solutions near interfaces) settings are taken into account.