An optimal estimate for the difference of solutions of two abstract evolution equations

Ikehata and Nishihara have established that the difference between any solution it of a linearly damped abstract wave equation and a certain solution v of a related abstract beat equation decays at least like t(-1)(log t)(1/2+epsilon) as time tends to infinity. They conjectured that the decay is in fact like t-1. We prove here the validity of this conjecture by relying on the spectral theorem for unbounded self-adjoint operators. We also establish the optimality of this estimate for the wave equation in an exterior domain. (C) 2003 Elsevier Science (USA). All rights reserved.