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

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 and Gérard (1999) [2] on R3, is performed by taking care of possible geometric effects. It uses some results of S. Ibrahim (2004) [21] on the behavior of concentrating waves on manifolds. © 2010 Elsevier Inc. All rights reserved.