Decay rates given by regularly varying functions for $C_0$-semigroups on Banach spaces
Genilson Santana, Silas L. Carvalho
TL;DR
The paper investigates decay rates of $C_0$-semigroups on Banach spaces when the generator's resolvent norm grows as a regularly varying function of index $\beta>0$. It extends prior polynomial and logarithmic decay results to the full class of regularly varying resolvent bounds, employing Fourier multiplier theory on Besov spaces and the sectorial functional calculus for complete Bernstein functions. The main contributions are precise decay estimates of the form $\|T(t)(1+A)^{-\tau}\| \lesssim t^{1-\frac{\tau-1/r}{\beta}}\big(\ell(t^{1/\beta})\big)^{\frac{\tau-1/r}{\beta}}$ when $\|( ext{Re}\lambda+A)^{-1}\| \lesssim (1+|\lambda|)^{\beta}\ell(1+|\lambda|)$ and analogous bounds involving $\kappa(\cdot)$ for the regularly varying case, with sharper (log-free) estimates in the Hilbert space setting. These results generalize the works of Deng–Rozendaal–Veraar and Santana–Carvalho by capturing arbitrary regularly varying growth and improving decay bounds for not-necessarily-bounded semigroups, thereby providing a broader, more versatile toolkit for analyzing long-time behavior of abstract evolution equations.
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 $β>0$, that is, as $|s|^β\ell(1+|s|)$ or $|s|^β/κ(1+|s|)$, where $\ell,κ$ 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 $|s|^β\log(1+|s|)^b$, with $b\ge 0$.