Abstract
We study the question of asymptotic stability, as time tends to infinity, of solutions of dissipative wave systems, governed by time-dependent nonlinear damping forces and by strongly nonlinear potential energies. This problem had been considered earlier for potential energies which 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 conclusions are related to and supplement earlier work of Payne and Sattinger [7], who treated the nondissipative case, and of Hale [1], who showed the existence of connected global attractors.
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