Linear stability of selfsimilar solutions of unstable thin-film equations

Abstract

We study the linear stability of selfsimilar solutions of long-wave unstable thin-film equations with power-law nonlinearities

Steady states, which exist for all values of $m$ and $n$ above, are shown to be stable if $m\le n+2$ when $0<n\le 2$, marginally stable if $m\le n+2$ when $2<n<3$, and unstable otherwise. Dynamical selfsimilar solutions are known to exist for a range of values of $n$ when $m=n+2$. We carry out the analysis of the stability of these solutions when $n=1$ and $m=3$. Spreading selfsimilar solutions are proven to be stable. Selfsimilar blowup solutions with a single local maximum are proven to be stable, while selfsimilar blowup solutions with more than one local maximum are shown to be unstable.

The equations above are gradient flows of a nonconvex energy on formal infinite-dimensional manifolds. In the special case $n=1$ the equations are gradient flows with respect to the Wasserstein metric. The geometric structure of the equations plays an important role in the analysis and provides a natural way to approach a family of linear stability problems.