Polynomial stability for a system of coupled strings

We study the long-time behavior of two vibrating strings which are coupled at a common boundary point by a damper. We show that the classical solutions converge polynomially with a uniform rate, where the decay exponent depends on number theoretic properties of the quotient of the wave speeds of the two springs. The proof is based on a resolvent characterization of polynomial stability due to Borichev–Tomilov and Batty–Duyckaerts.