Nonexistence of global weak solutions for a wave equation with nonlinear memory and damping terms

In this paper, we study the nonexistence of global weak solutions for a wave equation with nonlinear memory and damping terms. We give an answer to an open problem posed in [M. D’Abbicco, A wave equation with structural damping and nonlinear memory, Nonlinear Differ. Equ. Appl. 21 (2014), 751-773]. Moreover, comparing with the existing results, our results do not require any positivity condition of the initial values. The proof of our results is based on the asymptotic properties of solutions for an integral inequality.