Asymptotic profile and optimal decay of solutions of some wave equations with logarithmic damping

Abstract

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.