The main purpose of this paper is to study the properties of $\mathcal{P}\mathcal{R}$-semi-invariant submanifold of para-Kenmotsu manifold. We obtain the integrability conditions for the invariant distribution and anti-invariant distribution. We obtain some existence and non-existence results of $\mathcal{P}\mathcal{R}$-semi-invariant warped product submanifolds. We provide some necessary and sufficient conditions for $\mathcal{P}\mathcal{R}$-semi-invariant submanifold to be a $\mathcal{P}\mathcal{R}$-semi-invariant warped product submanifold in para-Kenmotsu manifold. We also derive some sharp inequalities for $\mathcal{P}\mathcal{R}$-semi-invariant warped product submanifold in para-Kenmotsu manifolds.

Our investigation involved sequentially warped product manifolds that contained gradient-normalized Ricci-harmonic solitons. We presented the primary connections for a gradient-normalized Ricci-harmonic soliton on sequential warped product manifolds. In practical applications, our research investigated gradient-normalized Ricci-harmonic solitons for sequential generalized Robertson-Walker spacetimes and sequential standard static space-times. Our finding generalized all results proven in ^[26].