Dynamic stability of non-minimizing phase mixtures

We study the dynamics of pattern formation in the one-dimensional partial differential equation u_tt - (W′(u_x))_x -u_xxt + u = 0 proposed recently by Ball, Holmes, Pego and Swart as a mathematical 'cartoon' for the dynamic formation of microstructures observed in various crystalline solids. Here W is a double-well potential like 1/4((u_x)² -1)². This problem possesses infinitely many equilibrium solutions, corresponding to different microstructures or phase mixtures, all locally energetically unstable in the natural norms. However, many are locally dynamically stable (theorem 1.1), and the paper proves this theorem and investigates other properties of these equilibria.