In this paper, we consider the following fourth-order hyperbolic equation involving variable exponents and supercritical damping: \(\begin{aligned} u_{tt}+\Delta ^{2} u-M(\Vert \nabla u\Vert _{2}^{2})\Delta u-\theta \Delta u_{t}+|u_{t}|^{m(x)-2}u_{t}=|u|^{p(x)-2}u, \end{aligned}\) where \(M(s)=a+b s^{r}\) is a positive \(C^{1}\) function with parameters \(a>0,b\ge 0,r>0,\theta \ge 0.\) Due to the failure of the embedding inequality for the supercritical case( \(m(x)>\frac{2N}{N-4}\) ), the well-known multiplier technique is unsuccessful in our problem. To end this, our strategy is to give a priori estimate for the weighted integral \(\textstyle \int _{\Omega }(2+t)^{1-m(x)}|u|^{m(x)}\textrm{d}x\) , and then to apply weighted multiplier method to prove that the energy functional decays logarithmically for supercritical damping. In particular, these results reveal that weak damping also effects the decay rate of the energy even if the strong damping is present. These results improved and extended the existing results obtained by Liao and Tan (Sci China Math. 66(2):285–302 2023).
