Time-Dependent Attractor for the Oscillon Equation

TL;DR

This paper studies the long-term behavior of solutions to a time-dependent Klein-Gordon equation in an expanding universe, proving the existence of a finite-dimensional, time-dependent global attractor.

Contribution

It introduces a novel dynamical framework to handle explicit time dependence and establishes the existence of a regular, finite-dimensional attractor for the nonautonomous system.

Findings

Existence of a regular, time-dependent global attractor.

Attractor sections have finite fractal dimension.

Framework applicable to equations with polynomial nonlinearities.

Abstract

We investigate the asymptotic behavior of the nonautonomous evolution problem generated by the Klein-Gordon equation in an expanding background, in one space dimension with periodic boundary conditions, with a nonlinear potential of arbitrary polynomial growth. After constructing a suitable dynamical framework to deal with the explicit time dependence of the energy of the solution, we establish the existence of a regular, time-dependent global attractor. The sections of the attractor at given times have finite fractal dimension.