<p>The purpose of this paper is to study the blow up phenomenon of the porous medium equation with a nonlocal integral term under nonlinear boundary conditions. We demonstrate that, under specified parameter conditions, the system undergoes finite-time blow up. By combining the maximum principles of the parabolic equation with differential inequality techniques, we estimate both upper and lower bounds for blow up time. Additionally, sufficient conditions for the system to avoid blowing up in a finite time are shown via a rigorous demonstration by contradiction. To illustrate these theoretical results, we present two concrete examples. Furthermore, we utilize the finite difference method to numerically simulate the equations. Graphical representations of the solutions are included to visually illustrate the blow up dynamics.</p>