Behavior of Kreiss bounded $C_0$-semigroups on a Hilbert space

TL;DR

This paper investigates the asymptotic behavior of $ ext{Kreiss}$ bounded $C_0$-semigroups on Hilbert spaces, establishing specific growth rates and applying results to perturbed wave equations.

Contribution

It proves a new asymptotic estimate for $ ext{Kreiss}$ bounded $C_0$-semigroups on Hilbert spaces and applies findings to perturbed wave equations.

Findings

Established $||T_t|| = O(t^{ ext{alpha}} / ootlog(t))$ growth rate

Provided application to perturbed wave equations

Extended understanding of semigroup asymptotics

Abstract

Let $0<\alpha \leq 1$. We prove that a $\alpha$-Kreiss bounded $C_0$ semigroup $(T_t)_{t \ge 0}$ on a Hilbert space has asymptotics $||T_t|| = O\big(t^{\alpha}/\sqrt{log(t)}\big)$. Then, we give an application to perturbed wave equation.