# Long-time dynamics of the strongly damped semilinear plate equation in $\mathbb{R}^{n}$

## TL;DR

This paper studies the long-term behavior of solutions to a damped semilinear plate equation in multi-dimensional space, proving the existence of a global attractor and its boundedness under certain damping conditions.

## Contribution

It establishes the existence and boundedness of a global attractor for the semilinear plate equation with localized damping in unbounded domains.

## Key findings

- Existence of a global attractor in H^2 × L^2 space.
- Boundedness of the attractor in higher regularity spaces.
- Conditions on damping coefficients ensure long-term stability.

## Abstract

We investigate the initial-value problem for the semilinear plate equation containing localized strong damping, localized weak damping and nonlocal nonlinearity. We prove that if nonnegative damping coefficients are strictly positive almost everywhere in the exterior of some ball and the sum of these coefficients is positive a.e. in $% \mathbb{R} ^{n}$, then the semigroup generated by the considered problem possesses a global attractor in $H^{2}\left( \mathbb{R} ^{n}\right) \times L^{2}\left( \mathbb{R} ^{n}\right) $. We also establish boundedness of this attractor in $ H^{3}\left( \mathbb{R} ^{n}\right) \times H^{2}\left( \mathbb{R} ^{n}\right) $.