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

Ikehata and Nishihara have established that the difference between any solution u of a linearly damped abstract wave equation and a certain solution v of a related abstract heat equation decays at least like 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.