Dynamics for a viscoelastic wave equation with nonlocal nonlinear dissipation and logarithmic nonlinearity: blow-up solutions, lifespan estimates and asymptotic stability

This paper investigates the instability of a class of wave equations with a non-local nonlinear damping term\begin{equation*} v_{tt}-\Delta v+(g\ast \Delta v)(t)+\sigma(\|\nabla v\|_{2}^{2})\phi(v_{t})=|v|^{p-2}v\ln|v|^{k},\end{equation*}where $(x,t)\in\Omega\times(0,T)$, $\Omega\subset \mathbb{R}^{n}$, $\sigma$ represents the nonlocal coefficient and $\phi$ is the nonlinear damping term. By considering suitable assumptions on the functions $\sigma$ and $\phi$, the exponents $p$ and $k$, the relaxation function $g$ and the initial data, and by making use of differentialinequality technique, we establish the occurrence of finite time blow up of solutions at low and arbitrary high positive initial energy levels. Moreover, lower bounds for the lifespan of solutions are derived in both cases. Asymptotic stability for the solution energy is also investigated by employing the energy perturbation method. This work extends and complementssome previous results in the literature.