Energy decay and blow-up of viscoelastic wave equations with polynomial nonlinearity and damping

This paper is concerned with the energy decay and the finite time blow-up of the solution to a viscoelastic wave equation with polynomial nonlinearity and weak damping. We establish explicit and general decay results for the solutions by imposing polynomial conditions on the relaxation function, provided that the initial energy is sufficiently small. Furthermore, we derive an upper bound for the blow-up time when the initial energy is less than the depth of the potential well by utilizing Levine’s convexity method. Additionally, we provide a lower bound for the blow-up time if the solution blows up.