Well-posedness, blow up and asymptotic stability for fractional pseudo-parabolic equation involving memory and logarithmic terms

Herein, we investigate the fractional pseudo-parabolic equation incorporating memory effect and logarithmic term $ \eta_{t}+(-\Delta)^s \eta+(-\Delta)^s \eta_{t} = \int_{0}^{t}k(t-\lambda)(-\Delta)^{s}\eta(\lambda)\,d\lambda+|\eta|^{r-2}\eta\ln|\eta| $ subject to Dirichlet boundary condition across various initial energy levels. To begin with, the local existence and uniqueness of weak solutions are rigorously established at any initial energy level via Galerkin approximation and contraction mapping principle. In addition, through the effective combination of Galerkin approximation, modified potential well theory, perturbed energy method and convexity theory etc, we explore the global existence and uniqueness, exponential and polynomial energy decay, as well as finite time blow up phenomena under low initial energy and critical initial energy levels.