  The paper gives a detailed study of long-time dynamics generated by weakly
damped wave equations in bounded 3D domains where the damping exponent depends
explicitly on time and may change sign. It is shown that in the case when the
non-linearity is superlinear, the considered equation remains dissipative if
the weighted mean value of the dissipation rate remains positive and that the
conditions of this type are not sufficient in the linear case. Two principally
different cases are considered. In the case when this mean is uniform (which
corresponds to deterministic dissipation rates), it is shown that the
considered system possesses smooth uniform attractors as well as non-autonomous
exponential attractors. In the case where the mean is not uniform (which
corresponds to the random dissipation rate, for instance, when this dissipation
rate is generated by the Bernoulli process), the tempered random attractor is
constructed. In contrast to the usual situation, this random attractor is
expected to have infinite Hausdorff and fractal dimension. The simplified model
example which demonstrates infinite-dimensionality of the random attractor is
also presented.
