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 failure of embedding inequality for supercritical case, well-known technique is unsuccessful in our problem. To do this, strategy give a priori estimate weighted integral $ \displaystyle }_{\Omega}(2+t)^{1-m(x)}|u|^{m(x)}{{{\text{d}}}} x and then apply modified multiplier method prove that energy functional decays logarithmically zero when relaxation function g exponentially zero. Meanwhile, more general cases, we also explicit dependence decay rate on both m(x) g. This differs from some results [21] where