Sharp polynomial decay rates for the damped wave equation on the torus.(English)Zbl 1295.35075

Summary: We address the decay rates of the energy for the damped wave equation when the damping coefficient $$b$$ does not satisfy the geometric control condition (GCC). First, we give a link with the controllability of the associated Schrödinger equation. We prove in an abstract setting that the observability of the Schrödinger equation implies that the solutions of the damped wave equation decay at least like $$1/\sqrt{t}$$ (which is a stronger rate than the general logarithmic one predicted by the Lebeau theorem). Second, we focus on the 2-dimensional torus. We prove that the best decay one can expect is $$1/t$$, as soon as the damping region does not satisfy GCC. Conversely, for smooth damping coefficients $$b$$ vanishing flatly enough, we show that the semigroup decays at least like $$1/t^{1-\epsilon}$$, for all $$\epsilon>0$$. The proof relies on a second microlocalization around trapped directions, and resolvent estimates. In the case where the damping coefficient is a characteristic function of a strip (hence discontinuous), Stéphane Nonnenmacher computes in an appendix part of the spectrum of the associated damped wave operator, proving that the semigroup cannot decay faster than $$1/t^{2/3}$$. In particular, our study emphasizes that the decay rate highly depends on the way $$b$$ vanishes.

MSC:

 35B40 Asymptotic behavior of solutions to PDEs 35A21 Singularity in context of PDEs 35B35 Stability in context of PDEs 35L05 Wave equation 35P20 Asymptotic distributions of eigenvalues in context of PDEs 35S05 Pseudodifferential operators as generalizations of partial differential operators 93C20 Control/observation systems governed by partial differential equations 58J45 Hyperbolic equations on manifolds 93B07 Observability
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