<p>This paper is concerned with the analysis of blow-up phenomena for the local mild solution of a nonlinear wave equation with past history. Such problems naturally arise in the study of viscoelasticity and nonlinear systems, particularly in modelling the longitudinal motion of viscoelastic materials governed by the Boltzmann law. Firstly, by employing the classical semigroup framework, the Lumer-Phillips theorem and the Lax-Milgram theorem, we establish the existence of local mild solutions. Subsequently, a refined argument is provided to derive the global existence of solutions. Furthermore, by means of the potential well method, we analyze the blow-up properties of local mild solutions under four distinct initial energy levels, namely, subcritical initial energy level, critical initial energy level, arbitrarily high initial energy level, and negative initial energy level. Additionally, the analytical techniques developed in this work can be extended to other classes of nonlocal problems with infinite memory arising in related fields.</p>