# On a fractional thin film equation

## TL;DR

This paper studies a fractional thin film equation of order between 2 and 4, proving existence of solutions, constructing explicit self-similar solutions, and exploring higher-order nonlocal equations with similar structures.

## Contribution

It introduces a fractional version of the thin film equation, establishes existence of solutions, and constructs explicit self-similar solutions for orders between 2 and 6.

## Key findings

- Existence of nonnegative weak solutions for fractional orders between 2 and 4.
- Explicit self-similar solutions with compact support.
- Extension to higher-order nonlocal equations with positive, compactly supported solutions.

## Abstract

This paper deals with a nonlinear degenerate parabolic equation of order $\alpha$ between 2 and 4 which is a kind of fractional version of the Thin Film Equation. Actually, this one corresponds to the limit value $\alpha=4$ while the Porous Medium Equation is the limit $\alpha=2$. We prove existence of a nonnegative weak solution for a general class of initial data, and establish its main properties. We also construct the special solutions in self-similar form which turn out to be explicit and compactly supported. As in the porous medium case, they are supposed to give the long time behaviour or the wide class of solutions. This last result is proved to be true under some assumptions. Lastly, we consider nonlocal equations with the same nonlinear structure but with order from 4 to 6. For these equations we construct self-similar solutions that are positive and compactly supported, thus contributing to the higher order theory.