Energy decay for a system of Schr{\"o}dinger equations in a wave guide

We could similarly consider the problem with damping and/or coupling on both sides of the boundary. This problem is completed with the initial conditions u|t“0 “ u0, v|t“0 “ v0, (1.3) where u0, v0 P L2pΩq. We will check that the problem (1.1)-(1.3) is well posed. If U “ pu, vq is a solution, then for t ě 0 we consider the energy Ept;Uq “ }uptq}2L2pΩq ` }vptq} 2 LpΩq . A straightforward computation shows that E is a non-increasing function of time: d dt Ept;Uq “ ́2a ż

[1]  J. Royer,et al.  Local energy decay for the damped wave equation , 2013, 1312.4483.

[2]  G. Lebeau,et al.  Stabilisation de l’équation des ondes par le bord , 1997 .

[3]  A. Matsumura,et al.  On the Asymptotic Behavior of Solutions of Semi-linear Wave Equations , 1976 .

[4]  J. Bouclet,et al.  Low Frequency Estimates and Local Energy Decay for Asymptotically Euclidean Laplacians , 2010, 1003.6016.

[5]  G. Lebeau,et al.  Equation des Ondes Amorties , 1996 .

[6]  S. Nonnenmacher,et al.  Sharp polynomial decay rates for the damped wave equation on the torus , 2012, 1210.6879.

[7]  Tosio Kato.  Perturbation theory for linear operators , 1966 .

[8]  L. Aloui,et al.  Stabilization of Schrödinger equation in exterior domains , 2007 .

[9]  J. Ralston.  Solutions of the wave equation with localized energy , 1969 .

[10]  Stabilization of trajectories for some weakly damped hyperbolic equations , 1985 .

[11]  Amru Hussein.  Maximal quasi-accretive Laplacians on finite metric graphs , 2012, 1211.4143.

[12]  Energy decay for damped wave equations on partially rectangular domains , 2006, math/0601195.

[13]  J. Royer.  Exponential Decay for the Schrödinger Equation on a Dissipative Waveguide , 2014, Annales Henri Poincaré.

[14]  J. Royer,et al.  Energy decay and diffusion phenomenon for the asymptotically periodic damped wave equation , 2017, Journal of the Mathematical Society of Japan.

[15]  R. Nagel,et al.  One-parameter semigroups for linear evolution equations , 1999 .

[16]  Jeffrey Rauch,et al.  Exponential Decay of Solutions to Hyperbolic Equations in Bounded Domains , 1974 .

[17]  L. Aloui,et al.  Stabilisation pour l'Équation des Ondes dans un Domaine Extérieur , 2002 .

[18]  J. Royer.  LOCAL DECAY FOR THE DAMPED WAVE EQUATION IN THE ENERGY SPACE , 2015, Journal of the Institute of Mathematics of Jussieu.

[19]  D. Krejčiřík,et al.  Non-self-adjoint graphs , 2013, 1308.4264.

[20]  C. Bardos,et al.  Microlocal ideas in control and stabilization , 1989 .

[21]  N. Burq.  Décroissance de l'énergie locale de l'équation des ondes pour le problème extérieur et absence de résonance au voisinage du réel , 1998 .

[22]  Anne Boutet de Monvel,et al.  Algebraic and Geometric Methods in Mathematical Physics , 1996 .

[23]  V. Edwards.  Scattering Theory , 1973, Nature.

[24]  Dietrich Hafner,et al.  Local Energy Decay for Several Evolution Equations on Asymptotically Euclidean Manifolds , 2010, 1008.2357.

[25]  Mark S. C. Reed,et al.  Method of Modern Mathematical Physics , 1972 .

[26]  J. Royer,et al.  Spectrum of a non-selfadjoint quantum star graph , 2019, 1911.04760.