The blow-up behavior of the solution for a higher-order nonlinear wave equation with source term

The main objective of this study is to analyze the behavior of weak local solutions to the high-order nonlinear wave equation with damping and source terms involving variable exponents with the homogeneous Dirichlet boundary conditions, which has significant applications in various fields, especially in physics. In this research: we initially presented some results previously proven by researchers, which we later utilized in our work. Considering assumptions on the variable exponents, we then established the existence and uniqueness of the solution using a combination of the fixed-point theory approach and the Galerkin method. In the last stage, we demonstrated that, with negative initial energy and by employing a Lyapunov function, the solution to the equation blows up in finite time.