<p>This paper focuses on a linear partial differential equation (PDE) with a fourth-order time derivative and high-dimensional spatial variables. Its high-dimensionality and higher-order properties pose challenges for investigating chaotic dynamics in this PDE. Taking the correlation of dynamical behaviors with coefficients as an entry point, we conduct a systematic exploration of the explicit relationship between chaotic behavior and the coefficient discriminant. Analysis demonstrates that when the coefficient discriminant is positive, this PDE exhibits Devaney chaos. Conversely, a non-positive discriminant indicates stability. This research approach enables the rapid determination of whether chaos exists through coefficients.</p>