On a Hartree-type nonlinearity wave equation with distributed delay combined with a fractional condition

This work focused on the analysis of a nonlinear wave equation of Hartree-type that includes a distributed delay term, where the delay effects are governed with fractional conditions. Such a formulation allows the model to incorporate long-range memory effects and anomalous dissipation phenomena, which are characteristic of complex media. The model captures complex memory and nonlocal interaction effects that arise in various physical systems, such as quantum mechanics and nonlinear optics. In particular, the fractional delay mechanism provides a more accurate description of hereditary effects than classical integer-order delay models. We worked under a framework that allows for initial data with negative energy and imposed suitable assumptions on the kernel functions and nonlinear terms. Using energy methods and a concavity argument, we rigorously proved that the solution to the system cannot exist globally in time and must blow up in finite time. Compared with the classical Hartree wave equation without delay or fractional effects, our results show that the combined presence of distributed delay and fractional damping significantly enhances the instability mechanism.