Resolvent estimates for the one-dimensional damped wave equation with unbounded damping

Abstract

We study the generator G of the one-dimensional damped wave equation with unbounded damping. We show that the norm of the corresponding resolvent operator, (G − λ) −1 , is approximately constant as |λ| → +∞ on vertical strips of bounded width contained in the closure of the left-hand side complex semi-plane, C − := {λ ∈ C : Re λ ≤ 0}. Our proof rests on a precise asymptotic analysis of the norm of the inverse of T (λ), the quadratic operator associated with G.