  We consider a wave equation with a nonlocal logarithmic damping depending on
a small parameter $\theta \in (0,1/2)$. This research is a counter part of that
was initiated by Charao-D'Abbicco-Ikehata considered in [5] for the large
parameter case $\theta \in (1/2,1)$. We study the Cauchy problem for this model
in the whole space for the small parameter case, and we obtain an asymptotic
profile and optimal estimates in time of solutions as time goes to infinity in
$L^2$-sense. An important discovery in this research is that in the one
dimensional case, we can present a threshold $\theta^{*} = 1/4$ of the
parameter $\theta$ such that the solution of the Cauchy problem decays with
some optimal rate for $\theta \in (0,\theta^{*})$, while the $L^2$-norm of the
corresponding solution blows up in infinite time for $\theta \in
[\theta^{*},1/2)$. The former (i.e., $\theta \in (0,\theta^{*})$ case)
indicates an usual diffusion phenomenon, while the latter (i.e., $\theta \in
[\theta^{*},1/2)$ case) implies, so to speak, a singular diffusion phenomenon.
Such a singular diffusion in the one dimensional case is a quite novel
phenomenon discovered through our new model produced by logarithmic damping
with a small parameter $\theta$.
