Asymptotic analysis of the abstract telegraph equation

Abstract

It is known that each solution of the telegraph equation \begin{equation} u^{\prime \prime}(t)+2au^{\prime}(t)+A^2u(t)=0, \tag*{(0.1)} \end{equation} $(A=A^* $on$\: \mathcal {H}, a>0)$ is approximately equal to some solution of the abstract heat equation, \begin{equation} 2av^\prime(t) + A^2v(t)=0. \tag*{(0.2)} \end{equation} It is shown how to find $v(0)$, in terms of $u(0)$ and $u^\prime(0)$, so that one can say that a given solution of (0.1) is like a specific solution of (0.2).