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In this article, we consider the initial-boundary value problem for a class of viscoelastic extensible beam equations with logarithmic source term, strong damping term, and weak damping term. We establish the local existence and uniqueness of weak solutions by applying the Faedo-Galerkin method and the contraction mapping principle. Under the framework of potential wells, for subcritical initial energy, we derive the sharp condition for initial data classification, i.e., the global existence and decay estimates of weak solutions in the stable set and the finite-time blow-up of weak solutions in the unstable set. Moreover, we derive sufficient conditions for the finite-time blow-up of weak solutions with negative initial energy and null initial energy. And we obtain an upper bound estimate for the blow-up time.

Qualitative properties of solutions to the viscoelastic beam equation with damping and logarithmic nonlinear source terms