Singular limit problem of abstract second order evolution equations

Abstract

We consider the singular limit problem in a real Hilbert space for abstract second order evolution equations with a parameter $\varepsilon \in (0,1]$. We first give an alternative proof of the celebrated results due to Kisynski (1963) from the viewpoint of the energy method. Next we derive a more precise asymptotic profile as $\varepsilon \to +0$ of the solution itself depending on $\varepsilon$ under rather high regularity assumptions on the initial data.