Linear stability of selfsimilar solutions of unstable thin-film equations
Dejan Slepčev · Interfaces and Free Boundaries Mathematical Analysis Computation and Applications · 2009
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≤ 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.