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

TL;DR

This paper studies the long-term behavior of weakly damped wave equations with time-varying, sign-changing damping, revealing conditions for dissipativity and the existence of various types of attractors, including deterministic and random ones.

Contribution

It introduces new conditions for dissipativity in wave equations with sign-changing damping and constructs both deterministic and random attractors, including cases with infinite-dimensional attractors.

Findings

Dissipativity depends on the weighted mean of damping being positive for superlinear non-linearity.

Existence of smooth uniform and non-autonomous exponential attractors in deterministic damping cases.

Construction of tempered random attractors with infinite Hausdorff and fractal dimension in stochastic damping scenarios.

Abstract

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.