Deterministic and random attractors for a wave equation with sign changing damping

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.