On the dynamics of fine structure

We investigate models for the dynamical behavior of mechanical systems that dissipate energy as time t increases. We focus on models whose underlying potential energy functions do not attain a minimum, possessing minimizing sequences with finer and finer structure that converge weakly to nonminimizing states. In Model 1 the evolution is governed by a nonlinear partial differential equation closely related to that of one-dimensional viscoelasticity, the underlying static problem being of mixed type. In Model 2 the equation of motion is an integro-partial differential equation obtained from that in Model 1 by an averaging of the nonlinear term; the corresponding potential energy is nonlocal. After establishing global existence and uniqueness results, we consider the longtime behavior of the systems. We find that the two systems differ dramatically. In Model 1, for no solution does the energy tend to its global minimum as t → ∞. In Model 2, however, a large, dense set of solutions realize global minimizing sequences; in this case we are able to characterize, asymptotically, how energy escapes to infinity in wavenumber space in a manner that depends upon the smoothness of initial data. We also briefly discuss a third model that shares the stationary solutions of the second but is a gradient dynamical system. The models were designed to provide insight into the dynamical development of finer and finer microstructure that is observed in certain material phase transformations. They are also of interest as examples of strongly dissipative, infinite-dimensional dynamical systems with infinitely many unstable "modes", the asymptotic fate of solutions exhibiting in the case of Model 2 an extreme sensitivity with respect to the initial data.