In this work, we study an initial boundary value problem for a quasi-linear parabolic system with a weak viscoelastic term of the form \begin{equation*} \begin{aligned} A(t) |u_{t}|^{m - 2} u_{t} +\Delta^{2}u - {\alpha}(t) &\int_{0}^{t}g( t - s) \Delta^{2}( s )ds = 0, \qquad x\in \Omega, t \geq 0, \\ u(x,t) = \frac {\partial u} {\partial \nu}(x,t)&= 0,\qquad x\in \partial \Omega,\ t \geq 0,\\ u( x , 0) = u_{0}&(x),\qquad x \in \Omega,\\ \end{aligned} \end{equation*} where $m\geq 2$, $A(t)$ a bounded and positive definite matrix is considered in a bounded domain $\Omega$ in $R^{n}$. We establish a general decay result, which depends on the behavior of both $\alpha$ and $g$, by using the perturbed energy functional technique.

In this paper we consider a viscoelastic double-Kirchhoff type wave equation of the form $$ u_{tt}-M_{1}(\|\nabla u\|^{2})\Delta u-M_{2}(\|\nabla u\|_{p(x)})\Delta_{p(x)}u+(g\ast\Delta u)(x,t)+\sigma(\|\nabla u\|^{2})h(u_{t})=\phi(u), $$ where the functions $M_{1},M_{2}$ and $\sigma, \phi$ are real valued functions and $(g\ast\nabla u)(x,t)$ is the viscoelastic term which are introduced later. Under appropriate conditions for the data and exponents, the general decay result and blow-up of solutions are proved with positive initial energy. This study extends and improves the previous results in the literature to viscoelastic double-Kirchhoff type equation with degenerate nonlocal damping and variable-exponent nonlinearities.