General decay for a system of viscoelastic wave equation with past history, distributed delay and Balakrishnan-Taylor damping terms

The subject of this research is a coupled system of nonlinear viscoelastic wave equations with distributed delay components, infinite memory and Balakrishnan-Taylor damping. Assume the kernels $g_i:{\mathbf{R}}_{+}\to {\mathbf{R}}_{+}$ holds true the below

$$g_i^{\prime }(t)\le -{\zeta }_i(t)G_i(g_i(t)),\mathrm{\forall }t\in {\mathbf{R}}_{+},\mathrm{f}\mathrm{o}\mathrm{r}i=1,2,$$

in which ${\zeta }_i$ and $G_i$ are functions. We demonstrate the stability of the system under this highly generic assumptions on the behaviour of $g_i$ at infinity and by dropping the boundedness assumptions in the historical data.