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

Abstract

We investigate the existence and the boundary regularity of source-type self-similar solutions to the thin-film equation $h_t=-(h^n{h}_{zzz}{)}_z+(h^{n+3}{)}_{zz},$ $t>0,z\in \mathbb{R};h(0,z)=\omega \delta (z)$ where $n\in (3/2,3),\omega >0$ 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: $h_t=-(h^n{h}_{zzz}{)}_z$. 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.