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

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.