Stabilizing Energy-Critical Wave Equation to a Finite Dimensional Attractor via Nonlinear Damping

TL;DR

This paper demonstrates that a 3D energy-critical wave equation with nonlinear damping can be stabilized to a finite-dimensional attractor, using advanced dissipation techniques and energy identities, under certain monotonicity conditions.

Contribution

It introduces a novel stabilization method for energy-critical wave equations with nonlinear damping, proving the existence of a finite-dimensional attractor and exponential attractor.

Findings

Global attractor exists for the system

Wave stabilization to a finite-dimensional set

Exponential attractor established

Abstract

The wave equation with energy critical sources and nonlinear damping defined on a 3D bounded domain is considered. It is shown that the resulting dynamical system admits a global attractor. Under the additional assumption of strong monotonicity of the damping at the origin, it is shown that the originally unstable quintic wave is uniformly stabilised to a finite dimensional and smooth set. Moreover, the existence of exponential attractor is established. In order to handle \enquote{energy criticality} of both sources and damping, the methods used depend on enhanced dissipation \cite{Bociu-lasiecka-jde}, energy {\it identity} for weak solutions \cite{Koch-lasiecka}, an adaptation of Ball's method \cite{ball}, and the theory of quasi-stable systems \cite{chueshov-white}.