Well-Posedness and Blow-Up of Solutions for a Variable Exponent Nonlinear Petrovsky Equation

In this article, we investigate a nonlinear Petrovsky equation with variable exponent and damping terms. First, we establish the local existence using the Faedo–Galerkin approximation method under the conditions of positive initial energy and appropriate constraints on the variable exponents and . Finally, we prove a finite-time blow-up result for negative initial energy.