Exponential decay and finite-time blow-up for the Klein–Gordon equation with linear strong damping, nonlinear weak damping, and logarithmic nonlinearity

This paper focuses on the initial boundary value problem for a class of Klein–Gordon equations that contains a linear strong damping term, a nonlinear weak damping term, and a nonlinear logarithmic term. Firstly, we establish the local existence and uniqueness of weak solutions by applying the Fadeo–Galerkin method. Secondly, we derive the global existence result of the weak solutions by the potential-well method in the case of subcritical initial energy, and exponential energy decay estimate result under certain conditions. Thirdly, we demonstrate the blow-up properties of weak solutions for low initial energy by applying differential inequality techniques, in addition to discussing the upper bound of the blow-up time. Finally, we extend the conclusions of global existence and energy decay in the case of critical initial energy.