Local asymptotic stability for nonlinear evolution equations with amplification forces

The asymptotic stability, as time tends to infinity, of solutions of dissipative wave systems is examined when time-dependent nonlinear damping forces are present and strongly nonlinear energies are present. The work extends previous studies in which the potential energies arose from restoring forces by also allowing for the effect of amplifying forces. On account of the amplifying forces, global asymptotic stability is no longer possible, and so must be replaced by local stability. After the main result has been proved the paper ends by presenting a number of illustrative examples and general remarks.