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

2010/04/16 by Camille Laurent, Laurent, Camille · 1 citation
Engineering · Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Nonlinear Photonic Systems #Stability and Controllability of Differential Equations #math.AP #math.OC

paper · pdf · doi:10.48550/arxiv.1004.2768

arxiv created 2010/04/16 · arxiv updated 2010/05/02

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-Gérard on \R³, is performed by taking care of possible geometric effects. It uses some results of S. Ibrahim on the behavior of concentrating waves on manifolds.

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