We study the question of asymptotic stability, as time tends to infini
ty, of solutions of dissipative wave systems, governed by time-depende
nt nonlinear damping forces and by strongly nonlinear potential energi
es. This problem had been considered earlier for potential energies wh
ich arise from restoring forces, whereas here we allow as well for the
effect of amplifying forces. Global asymptotic stability can then no
longer be expected, and should be replaced by local stability. The con
clusions are related to and supplement earlier work of Payne and Satti
nger [7], who treated the nondissipative case, and of Hale [1], who sh
owed the existence of connected global attractors.