Global attractors for the Signorini problem with pointwise damping

TL;DR

This paper proves the existence of compact global attractors and exponential decay in the Signorini problem with pointwise damping, using a hybrid PDE-ODE approximation for analysis.

Contribution

It introduces a novel hybrid PDE-ODE approach to analyze long-term behavior and attractors in Signorini problems with damping.

Findings

Solutions decay exponentially to zero.

Existence of compact global attractors.

Applicable to elastic obstacle problems with normal compliance.

Abstract

The existence of global attractors is investigated for the Signorini problem with pointwise dissipation. It is shown that both the semilinear Signorini problem and the elastic obstacle problem with normal compliance exhibit exponential decay to zero and admit compact global attractors. To establish these results, the original problem is approximated by a hybrid PDE-ODE system, which allows for a rigorous analysis of well-posedness and the long-time behavior of its solutions.