Abstract
We introduce a new model of the logarithmic-type of wave-like equation with a nonlocal logarithmic damping mechanism, which is weakly effective compared with the frequently studied fractional damping equations. We consider the Cauchy problem for this new model in Rn, and study the asymptotic profile, optimal decay, and/or blow-up rates of the solutions as t→∞ in the L2-sense. The L operator considered in this study was used to dissipate the solutions of the wave equation investigated by Charão-Ikehata [6] and in the low frequency parameters, the principal part of the equation and the damping term were rather weakly effective compared with the widely studied power-type equations, such as (−Δ)θut with θ∈(0,1].
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