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

Irena Lasiecka, Vando Narciso

TL;DR

This work analyzes a 3D energy-critical wave equation with nonlinear damping and quintic source on a bounded domain. By employing enhanced dissipation, energy identities for weak solutions, Ball-type absorbing arguments, and quasi-stability, the authors establish the existence of a global attractor that is finite-dimensional and smooth, as well as an exponential attractor. They prove Hadamard well-posedness, asymptotic compactness, and a gradient structure, and further show maximal regularity of the attractor, including higher-order spatial regularity and backward-in-time smoothness. The results significantly advance understanding of long-time dynamics for energy-critical hyperbolic equations with nonlinear damping, enabling finite-dimensional reductions and potential data-assimilation approaches for practical computation and control.

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}.

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

TL;DR

This work analyzes a 3D energy-critical wave equation with nonlinear damping and quintic source on a bounded domain. By employing enhanced dissipation, energy identities for weak solutions, Ball-type absorbing arguments, and quasi-stability, the authors establish the existence of a global attractor that is finite-dimensional and smooth, as well as an exponential attractor. They prove Hadamard well-posedness, asymptotic compactness, and a gradient structure, and further show maximal regularity of the attractor, including higher-order spatial regularity and backward-in-time smoothness. The results significantly advance understanding of long-time dynamics for energy-critical hyperbolic equations with nonlinear damping, enabling finite-dimensional reductions and potential data-assimilation approaches for practical computation and control.

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}.
Paper Structure (26 sections, 12 theorems, 175 equations)

This paper contains 26 sections, 12 theorems, 175 equations.

Table of Contents

  1. Introduction
  2. Functional Setting, Assumptions and Main Results
  3. Functional Setting and the Assumptions Imposed
  4. Statements of the Main Results
  5. Past contributions-history of the problem.
  6. Proof of Theorem \ref{['theo-global']}: Weak Solutions and the Generation of the Dynamical System $(\mathcal{H},S_t)$.
  7. Proof of Theorem \ref{['theo-global']}-(I): Construction of Weak Solutions
  8. Analysis of linear terms.
  9. Analysis of the source term $f(u^k)$.
  10. Analysis of the damping term $g(u^k_t)$.
  11. Proof of Theorem \ref{['theo-global']}-(II): Energy Equality
  12. Proof of Theorem \ref{['theo-global']}-(III): Hadamard Well-posedness
  13. Generation of a Dynamical System
  14. Proof of Theorem \ref{['theo_main1']}: Global Attractors
  15. Asymptotic Compactness
  16. ...and 11 more sections

Key Result

Theorem 2.2

In reference to problem P, under Assumption Assumption, the following statements hold.

Theorems & Definitions (25)

  • Remark 2.1
  • Definition 2.1
  • Theorem 2.2
  • Remark 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Theorem 2.6
  • Remark 3.1
  • Proposition 3.2
  • Remark 3.3
  • ...and 15 more