Existence and regularity of source-type self-similar solutions for stable thin-film equations

TL;DR

This paper proves the existence and boundary regularity of source-type self-similar solutions for a class of thin-film equations, showing higher order corrections are analytic and depend on irrational powers, extending previous asymptotic results.

Contribution

It establishes the analyticity of higher order corrections in self-similar solutions for the thin-film equation, refining known asymptotic expansions near the support edge.

Findings

Higher order corrections are analytic functions of spatial variables.

Corrections depend on irrational powers, except when n=2.

Supports the asymptotic behavior matching traveling-wave solutions.

Abstract

We investigate the existence and the boundary regularity of source-type self-similar solutions to the thin-film equation $h_t=-(h^nh_{zzz})_z+(h^{n+3})_{zz},$ $ t>0,\; z\in \mathbb{R};\; h(0,z)= \omega \delta(z)$ where $n\in (\frac{3}{2},3),\; \omega > 0$ and $\delta$ is the Dirac mass at the origin. It is known that the leading order expansion near the edge of the support coincides with that of a traveling-wave solution for the standard thin-film equation: $h_t=-(h^nh_{zzz})_z$. In this paper we sharpen this result, proving that the higher order corrections are analytic with respect to three variables: the first one is just the {spatial} variable, whereas the second and the third (except for $n = 2$) are irrational powers of it. It is known that this third variable does not appear for the thin-film equation without gravity.