# Linear stability of selfsimilar solutions of unstable thin-ﬁlm equations

### Dejan Slepčev

Carnegie Mellon University, Pittsburgh, United States

## Abstract

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

*ut* = −(*un uxxx* + *um ux*)*x* for 0 < *n* < 3, *n* *m*.

Steady states, which exist for all values of m and n above, are shown to be stable if *m* ≤ *n* + 2 when 0 < *n* ≤ 2, marginally stable if *m* ≤ *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.

## Cite this article

Dejan Slepčev, Linear stability of selfsimilar solutions of unstable thin-ﬁlm equations. Interfaces Free Bound. 11 (2009), no. 3, pp. 375–398

DOI 10.4171/IFB/215