  We introduce a new model of the logarithmic type of wave like plate equation
with a nonlocal logarithmic damping mechanism. We consider the Cauchy problem
for this new model in the whole space, and study the asymptotic profile and
optimal decay rates of solutions as time goes to infinity in L^{2}-sense. The
operator L considered in this paper was first introduced to dissipate the
solutions of the wave equation in the paper studied by Charao-Ikehata in 2020.
We will discuss the asymptotic property of the solution as time goes to
infinity to our Cauchy problem, and in particular, we classify the property of
the solutions into three parts from the viewpoint of regularity of the initial
data, that is, diffusion-like, wave-like, and both of them.
