The current paper discusses the global existence and asymptotic behavior of solutions of the following new nonlocal problem$$ u_{tt}-M\left(\displaystyle \int_{\Omega}|\nabla u|^{2}\, dx\right)\triangle u + \delta u_{t}= |u|^{\rho-2}u \log|u|, \quad \text{in}\ \Omega \times ]0, \infty[, $$where\begin{equation*}M(s)=\left\{\begin{array}{ll}{a-bs, }&{\text{for}\ s \in [0, \frac{a}{b}[, }\\{0, }&{\text{for}\ s \in [\frac{a}{b}, +\infty[. }\end{array}\right. \end{equation*}If the initial data are appropriately small, we derive existence of global strong solutions and the exponential decay of the energy.