Global Existence and Blow‐Up Phenomenon of Solution for a Viscoelastic Wave Equation With Nonlinear Degenerate Damping and Logarithmic Source Term

ABSTRACT This paper deals with the initial boundary value problem for degenerate damped nonlinear wave equations with logarithmic nonlinearity, given by the equation: in a bounded region with the smooth boundary . We prove the existence and uniqueness of the weak solution through a combination of constructing an auxiliary problem, employing the Faedo–Galerkin method, and applying the contraction mapping principle under suitable conditions. Subsequently, we establish the existence of a global solution and derive the finite‐time blow‐up results for cases with negative initial energy. Moreover, we propose new conditions for the global solution and demonstrate that the weak solution will blow up in finite time when the initial energy is positive, by considering potential well theory and constructing Lyapunov functions.