Local decay of $C_0$-semigroups with a possible singularity of logarithmic type at zero

We prove decay rates for a vector-valued function $f$ of a non-negative real variable with bounded weak derivative, under rather general conditions on the Laplace transform $\hat{f}$. This generalizes results of Batty-Duyckaerts (2008) and other authors in later publications. Besides the possibility of $\hat{f}$ having a singularity of logarithmic type at zero, one novelty in our paper is that we assume $\hat{f}$ to extend to a domain to the left of the imaginary axis, depending on a non-decreasing function $M$ and satisfying a growth assumption with respect to a different non-decreasing function $K$. The decay rate is expressed in terms of $M$ and $K$. We prove that the obtained decay rates are essentially optimal for a very large class of functions $M$ and $K$. Finally we explain in detail how our main result improves known decay rates for the local energy of waves on exterior domains.