On stabilization and control for the critical Klein-Gordon equation on a 3-D compact manifold

TL;DR

This paper investigates the stabilization and control of the critical nonlinear Klein-Gordon equation on 3-D compact manifolds, establishing exponential decay under specific geometric conditions using advanced microlocal and profile decomposition techniques.

Contribution

It introduces a new geometric assumption slightly stronger than the classical condition and proves exponential decay for bounded solutions, extending control theory to complex geometric settings.

Findings

Proved exponential decay under new geometric assumptions.

Developed a profile decomposition adapted to manifold geometry.

Integrated microlocal analysis with geometric effects for stabilization.

Abstract

In this article, we study the internal stabilization and control of the critical nonlinear Klein-Gordon equation on 3-D compact manifolds. Under a geometric assumption slightly stronger than the classical geometric control condition, we prove exponential decay for some solutions bounded in the energy space but small in a lower norm. The proof combines profile decomposition and microlocal arguments. This profile decomposition, analogous to the one of Bahouri-G\'erard on $\R^3$, is performed by taking care of possible geometric effects. It uses some results of S. Ibrahim on the behavior of concentrating waves on manifolds.