Eigenmodes of the damped wave equation and small hyperbolic subsets

Abstract

We study stationary solutions of the damped wave equation on a compact and smooth Riemannian manifold without boundary. In the high frequency limit, we prove that a sequence of $\beta$-damped stationary solutions cannot be completely concentrated in small neighborhoods of a small fixed hyperbolic subset made of $\beta$-damped trajectories of the geodesic flow. The article also includes an appendix (by S. Nonnenmacher and the author) where we establish the existence of an inverse logarithmic strip without eigenvalues below the real axis, under a pressure condition on the set of undamped trajectories.

24 pages. With an appendix by S. Nonnenmacher and the author. In this new version, we modified an uncorrect exponent in the statement of Theorem A.1 from the appendix

References in corpus (2)

• Energy decay for the damped wave equation under a pressure condition
• From resolvent estimates to damped waves

Cited by in corpus (5)

• Resonances in open quantum maps
• Semi-uniform stability of operator semigroups and energy decay of damped waves
• Concentration Of Laplace Eigenfunctions And Stabilization Of Weakly Damped Wave Equation
• Semiclassical asymptotics for nonselfadjoint harmonic oscillators
• Semiclassical behaviour of quantum eigenstates