GLOBAL EXISTENCE AND BLOW-UP PHENOMENA FOR A HYPERBOLIC P-LAPLACIAN EQUATION WITH LOGARITHMIC NONLINEARITY AND NONLINEAR BOUNDARY CONDITIONS

In this paper, we investigate the global existence and finite-time blow-up phenomena for a class of hyperbolic equations involving nonlinear $p$-Laplacian diffusion, damping effects, logarithmic source terms, and nonlinear boundary conditions. The considered model is governed by

$u_{tt}-\operatorname{div}\big(|\nabla u^m|^{p-2}\nabla u^m\big)+\mu u_t= k(t)\, u\ln(1+u), \quad (x, t)\in\Omega\times(0, T), $

subject to nonlinear boundary conditions and suitable initial data, where $\Omega\subset\mathbb{R}^n$ $(n\ge2)$ is a bounded domain with smooth boundary, $p\ge2$, $m\ge1$, $\mu>0$, and $k(t)$ is a nonnegative differentiable function. By constructing suitable energy and auxiliary functionals, we establish sufficient conditions for the global existence of weak solutions. Moreover, under appropriate assumptions on the logarithmic source term and boundary nonlinearity, we prove that solutions blow up in finite time $T^\ast < \infty$ and derive explicit upper and lower estimates for the blow-up time. In addition, comprehensive numerical simulations are presented to illustrate the theoretical results and to describe the influence of the damping coefficient, logarithmic source intensity, and initial energy on the qualitative behavior of solutions. The computations confirm the sharpness of the analytical criteria and highlight the delicate interplay between hyperbolic dynamics, nonlinear diffusion, damping mechanisms, and logarithmic nonlinearities. The obtained results extend and complement several recent studies devoted to nonlinear parabolic and pseudo-parabolic equations involving $p$-Laplacian operators and logarithmic nonlinearities.