Double diffusion structure of logarithmically damped wave equations with a small parameter

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 $\theta$.