We consider the second order Cauchy problem $\varepsilon u_{\varepsilon}''+|A^{1/2}u_{\varepsilon}|^{2\gamma}Au_{\varepsilon}+u_{\varepsilon}'=0$, $u_{\varepsilon}(0)=u_0\neq 0$, $u_{\varepsilon}'(0)=u_1$ where $\varepsilon>0$, $H$ is a Hilbert space, $A$ is a self-adjoint positive operator on $H$ with dense domain $D(A)$, $(u_{0},u_{1})\in D(A)\times D(A^{1/2})$, and $\gamma > 0$. We accurately study the decay as $t$ goes to infinity of the solutions, which exist for every $\varepsilon$ small enough. In particular we obtain a new estimate on $u_{\varepsilon}''(t)$ and we show that $(1+t)^{1/2\gamma} Au_{\varepsilon}(t) \rightarrow u_{\varepsilon,\infty} \neq 0$, $(1+t)^{1+1/2\gamma} A^{1/2}u_{\varepsilon}'(t) \rightarrow v_{\varepsilon,\infty} \neq 0,$ as $t$ goes to infinity. Moreover we show that the norm of $u_{\varepsilon,\infty}$ and $v_{\varepsilon,\infty}$ does not depend on the initial data.