Linear stability of selfsimilar solutions of unstable thin-film equations

Abstract

Steady states, which exist for all values ofm andn above, are shown to be stable ifm 6nC 2 when 0 < n 6 2, marginally stable if m 6 nC 2 when 2 < n < 3, and unstable otherwise. Dynamical selfsimilar solutions are known to exist for a range of values ofn whenmDnC 2. We carry out the analysis of the stability of these solutions when nD 1 and mD 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 casenD 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. 2000 Mathematics Subject Classification: 35B35, 35K55, 76A20.