Abstract damped wave equations: The optimal decay rate

The exponential decay rate of the semigroup $S(t)=e^{t\mathbb{A}}$ generated by the abstract damped wave equation $$\ddot u + 2f(A) \dot u +A u=0$$ is here addressed, where $A$ is a strictly positive operator. The continuous function $f$, defined on the spectrum of $A$, is subject to the constraints $$\inf f(s)>0\qquad\text{and}\qquad \sup f(s)/s <\infty$$ which are known to be necessary and sufficient for exponential stability to occur. We prove that the operator norm of the semigroup fulfills the estimate $$\|S(t)\|\leq Ce^{\sigma_*t}$$ being $\sigma_*<0$ the supremum of the real part of the spectrum of $\mathbb{A}$. This estimate always holds except in the resonant cases, where the negative exponential $e^{\sigma_*t}$ turns out to be penalized by a factor $(1+t)$. The decay rate is the best possible allowed by the theory.