This paper investigates a p-Laplace higher-order hyperbolic equation with strong and weak damping terms and a superlinear source: \(\begin{aligned} u_{tt}-\Delta _{p}u+\Delta ^{2}u-\Delta u_{t}+|u_{t}|^{m-1}u_{t}=|u|^{q-1}u, \end{aligned}\) in \(\Omega \times (0,T_{\textrm{max}})\) , subject to null Navier boundary conditions. Here, \(\Omega \subset {\mathbb {R}}^n\) is a bounded open domain. By using the Banach contraction mapping principle, we establish the well-posedness of weak solutions. When \(q + 1 \le p\) , we prove that all the weak solutions remain globally bounded. For \(q + 1 > p\) , within the potential well framework, we derive the global existence of solutions for both critical and subcritical initial energy cases, accompanied by distinct decay estimates for global solutions when \(q+1 > p\) , initial energy \(E(0) \le d\) and Nehari functional \(I(u_0)\ge 0\) . Additionally, under specific exponent conditions (e.g., \(2\le m+1< p < q+1\) for negative initial energy, \(\max \{p, m+1\}<q+1\) for non-negative initial energy), we characterize finite-time blow-up of solutions under both positive and negative initial energy conditions. Using an auxiliary function method, we further demonstrate finite-time blow-up for linear weak damping with subcritical initial energy, and derive the bounds for the blow-up time.
