Stability Results for Higher Order Solutions of Damped Wave Equations With Generalized Hartree-Type Nonlinearity in Rn

In this paper, we investigate the dynamics of higher-order solutions for a class of damped wave equations posed in Rn and driven by a nonlocal cubic convolution source of Hartree type. The model incorporates a higher order Laplacian of order σ, spatially dependent density functions, and frictional damping mechanisms. Using energy methods combined with Hardy–Littlewood–Sobolev inequalities and higher order Sobolev embeddings, we establish the local well-posedness of solutions in suitable high-regularity spaces. We prove the global existence of solutions in the subcritical regime relative to the higher order Sobolev embedding and derive polynomial decay rates for the associated energy functional. The decay behavior is shown to depend explicitly on the order of the higher order operator and the singularity exponent of the convolution kernel.