Abstract
We study the long-time dynamics of a strongly damped $ p $-Laplacian wave equation posed on bounded domains. While the potential well method is a classical tool for semilinear and weakly damped equations, its extension to quasilinear problems with strong damping has remained less developed. In this work, we construct a one-parameter family of potential wells that organizes the solution behavior and provides a sharp threshold between global existence and finite-time blow-up. Specifically, for initial energies below the mountain-pass level, the data split into two invariant sets determined by the Nehari functional: one yields global weak solutions, while the other forces blow-up, with an intermediate vacuum region that no trajectory can enter. At the critical energy level, we prove analogous dichotomies under mild sign assumptions. Our framework extends and unifies potential well theory in the quasilinear, strongly damped setting, and applies to standard power-type nonlinearities.