Summary: We study the linear stability of selfsimilar solutions of long-wave unstable thin-film equations with power-law nonlinearities \[ u_{t} = - (u^{n} u_{xxx} + u^{m} u_{x})_x \qquad \text{for }0 < n < 3, n \leqslant m. \] Steady states, which exist for all values of \(m\) and \(n\) above, are shown to be stable if \(m \leqslant n + 2\) when \(0 < n \leqslant 2\), marginally stable if \(m\leqslant n + 2\) 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.