A note on the fourth-order hyperbolic equation with distinct dissipation terms and variable-exponent damping

In this paper, we investigate the properties of solutions to an initial-boundary value problem for a class of forth-order hyperbolic equations with the variable-exponent damping and a supercritical source. First, leveraging Galerkin's method and Banach's fixed-point theorem, we establish the well-posedness of the weak solution. Within the framework of potential well theory, and in conjunction with dispersion-dissipation terms of varying intensities, we prove the global existence of weak solutions and derive distinct energy decay estimates. It is shown that the linear weak damping yields an exponential decay rate, while the nonlinear weak damping induces polynomial decay rate. Furthermore, we classify the scenarios where weak solutions blow up in finite time, and further obtain upper and lower bounds for blow-up time.