  We introduce a new model of the logarithmic type of wave-like equation with a
nonlocal logarithmic damping mechanism, which is rather weakly effective as
compared with frequently studied fractional damping cases. We consider the
Cauchy problem for this new model in the whole space, and study the asymptotic
profile and optimal decay and/or blowup rates of solutions as time goes to
infinity in L^{2}-sense. The operator L considered in this paper was used to
dissipate the solutions of the wave equation in the paper studied by
Charao-Ikehata in 2020, and in the low frequency parameters the principal part
of the equation and the damping term is rather weakly effective than those of
well-studied power type operators.
