Abstract:
The paper gives a detailed study of long-time dynamics generated by
weakly damped wave equations in bounded 3D domains where the damping
coefficient depends explicitly on time and may change sign. It is shown that
in the case, where 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 rate), 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 dimensions. The simplified model example
demonstrating infinite-dimensionality of the random attractor is also
presented.