An effect of decay rates: Supercritical damping and a viscoelastic term

We consider the following viscoelastic equation involving variable exponent nonlinearities:$ u_{tt}-\Delta u+\displaystyle {\int }_0^tg(t-s)\Delta u(s){{{\text{d}}}} s+a|u_{t}|^{m(x)-2}u_{t} = |u|^{q(x)-2}u. $Due to the failure of the embedding inequality for the supercritical case, the well-known technique is unsuccessful in our problem. To do this, our strategy is to give a priori estimate for the weighted integral $ \displaystyle {\int }_{\Omega}(2+t)^{1-m(x)}|u|^{m(x)}{{{\text{d}}}} x $ and then apply modified multiplier method to prove that the energy functional decays logarithmically to zero when the relaxation function $ g $ decays exponentially to zero. Meanwhile, for more general cases, we also give the explicit dependence of decay rate on both the exponent $ m(x) $ and the relaxation function $ g. $ This differs from some results of [21] where the energy functional decays exponentially to zero when the relaxation function $ g $ decays exponentially to zero.