Well‐posedness for a class of wave equations with nonlocal weak damping

In this paper, we investigate the boundary value problem of the wave equation with nonlocal damping and nonlinear source term. The main purpose of this paper is to provide a systematic research on the dynamic behavior of the solutions with three different energy levels. More precisely, we prove the global existence, energy decay estimate and blow‐up of solution at both subcritical ( ) and critical ( ) initial energy levels. We also prove the finite time blow‐up of the solution for the initial data at arbitrary high energy level, including the estimates of lower and upper bounds of the blow‐up time.