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

We consider a wave equation with nonlocal logarithmic damping depending on small parameter 0 < ? 1 2 . This research is counter part of that was initiated by Charão-D'Abbicco-Ikehata considered in [5] for the large case > study Cauchy problem this model R n to ? ( , ) and we obtain an asymptotic profile optimal estimates time solutions as t ? ? L -sense. An important discovery when = can present threshold ? 4 such solution decays some rate while -norm corresponding never [ and, particular, it shows infinite blow-up solutions. The former (i.e., case) indicates usual diffusion phenomenon, latter implies, so speak, singular phenomenon. Such one dimensional quite novel phenomenon discovered through our new produced It might be already prepared structural ? ? u / however unfortunately nobody has ever just pointed out even case.