We introduce a new model of the nonlocal wave equation with a logarithmic damping mechanism, which is rather weak as compared with frequently studied fractional damping cases. We consider the Cauchy problem for the new model in \(\mathbf{R}^{n}\) and study the asymptotic profile and optimal decay rates of solutions as \(t \rightarrow \infty \) in \(L^{2}\)-sense. The damping terms considered in this paper is not studied so far, and in the low-frequency parameters, the damping is rather weakly effective than that of well-studied power type one such as \((-\Delta )^{\theta }u_{t}\) with \(\theta \in (0,1)\). When getting the optimal rate of decay, we meet the so-called hypergeometric functions with special parameters, so the analysis seems to be more difficult and attractive.

我们引入了具有对数阻尼机制的非局部波动方程的新模型，与经常研究的分数阻尼情况相比，该模型较弱。我们考虑\（\ mathbf {R} ^ {n} \）中新模型的柯西问题，并以\（t \ rightarrow \ infty \）在\（L ^ { 2} \）-感觉。到目前为止，尚未研究本文中考虑的阻尼项，并且在低频参数中，该阻尼的效果比经过深入研究的功率类型之一（\（（-\ Delta）^ {\ theta} u_ {t} \）与\（\ theta \ in（0,1）\）。当获得最佳衰减率时，我们会遇到带有特殊参数的所谓超几何函数，因此分析似乎更加困难且有吸引力。