Blow-up phenomena of boundary delay effects in nonlinear logarithmic wave dynamics

This study investigates wave equation in confined and smooth domain $ \Omega \subset \mathbb{R}^n $ that incorporates a nonlinear logarithmic source term with the time-varying delay and a frictional damping at the boundary.The local existence result has been examined by using semi-group theory, where as global existence of the solution has been investigated using the energy approach. To further enhance the significance of this work, the blow-up phenomenon of this model is demonstrated numerically to support the theoretical analysis when the initial energy is negative.