Blow-Up Dynamics of Solutions to a Nonlinear Wave Equation with Positive Initial Energy

This paper investigates the dynamics of a quasi-linear partial differential equation of fourth order characterized by bi-hyperbolic properties, incorporating dynamic boundary conditions.The study focuses on the interplay between the equation's nonlinear source term, the boundary effects, and the initial energy.By applying the concavity method, we derive conditions that lead to the finitetime blow-up phenomenon in solutions with non-negative initial energy.These findings highlight the impact of dynamic boundary conditions on the development of finite-time singularities in higher-order hyperbolic equations.