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