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

Abstract

RésuméOn étudie la stabilisation et le contrôle interne de l'équation de KleinGordon critique sur des variétés de dimension 3. Sous des conditions géomé-triques légèrement plus fortes que la condition de contrôle géométrique classique, on prouve la décroissance exponentielle de solutions bornées dans l'espace d'énergie mais petites dans des normes plus faibles. La preuve combine la décomposition en profils et des arguments microlocaux. Cette décomposition, analogue à celle de Bahouri-Gérard [2] sur R 3 , nécessite l'analyse de certains effets dus à la géométrie. Elle utilise des résultats de S. Ibrahim [16] sur le comportement d'ondes de concentration sur les variétés. AbstractWe 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 [2] on R 3 , is performed by taking care of possible geometric effects. It uses some results of S. Ibrahim [16] on the behavior of concentrating waves on manifolds.

On the global well-posedness of energy-critical Schrödinger equations in curved spaces

“…Other critical dispersive models, such as large-data critical wave equations or the Klein-Gordon equation have also been studied extensively, both in the case of the Minkowski space and in other Lorentz manifolds. See for example [3,4,16,17,27,28,34,35,36,38,41,43,44,48,49,52] and the book [54] for further discussion and references. In the case of the wave equation, passing to the variable coefficient setting is somewhat easier due the finite speed of propagation of solutions.…”

Decay rate estimates for the wave equation with subcritical semilinearities and locally distributed nonlinear dissipation

“…The case of the semilinear wave equation as in the present article also attracted the attention of many researchers. However, most results assume that the damping is either linear (see e.g., [20], [21], [29], [31], [38], and [45]) or linearly bounded [10]. In the present article, we generalize the known results mentioned on the subject to a considerably larger class of dissipative effects that are not necessarily linear or linearly bounded.…”

A Note on the Semiglobal Controllability of the Semilinear Wave Equation

“…The proof is mutatis mutandis given in [32]. See also [8], Theorem 3 of [10], Theorem 3.2 of [21] for control close to 0. See also [22] where local control near trajectories are constructed for the nonlinear Schrödinger equation.…”