Decay rates given by regularly varying functions for $C_0$-semigroups on Banach spaces

TL;DR

This paper investigates decay rates of $C_0$-semigroups on Banach spaces, establishing new estimates when the resolvent norm grows as a regularly varying function, extending and improving previous results.

Contribution

It extends existing decay rate estimates to cases where the resolvent norm exhibits regularly varying growth, broadening the applicability of decay analysis for $C_0$-semigroups.

Findings

Extended decay estimates for regularly varying resolvent growth

Improved bounds for logarithmic growth cases

Generalized results beyond polynomial growth

Abstract

We study rates of decay for (possibly unbounded) $C_0$-semigroups on Banach spaces under the assumption that the norm of the resolvent of the respective semigroup generator grows as a regularly varying function of type $\beta>0$, that is, as $|s|^{\beta}\ell(1+|s|)$ or $|s|^{\beta}/\kappa(1+|s|)$, where $\ell,\kappa$ are arbitrary monotone and slowly varying functions. The main result extends the estimates obtained by Deng, Rozendaal and Veraar (J. Evol. Equ. 24, 99 (2024)) to this setting of regularly varying functions and improves the estimates obtained by Santana and Carvalho (J. Evol. Equ. 24, 28 (2024)) in case $|s|^{\beta}\log(1+|s|)^b$, with $b\ge 0$.