In this work, we investigate the logarithmic nonlinear wave equation, which is distinguished by strong time-varying delay components, infinite memory, and strong damping. Through semigroup theory, we have established a local existence result, paving the way for a comprehensive understanding of the equation’s behavior. In addition, we explore the global existence of solutions, revealing fascinating decay results and investigating the circumstances in which blow-up may occur in solutions with negative starting energy. Our results shed light on the complex nature of this dynamic system while also strengthening the theoretical framework.

In this article, we consider the dynamics of a viscoelastic plate equation with internal fractional damping, a nonlinear logarithmic source, and infinite memory effects. The existence of a local weak solution is shown effectively through the framework of semigroup theory. Furthermore, we show that the blow-up in finite time of the local solution may occur under specific conditions and is demonstrated within the development of a suitable Lyapunov functional. Our result offers an insight into the challenges presented by this class of equations and their relevance to physical systems.

This study examines a nonlinear wave equation with boundary fractional damping and time-varying delay. After proving the existence of a global solution, we establish exponential stability by constructing an appropriate Lyapunov functional. Finally, we prove the blow-up phenomenon using the potential-well method.

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.