There are several techniques in the classical case for some integro-differential equations involving the concept of entropy to show some properties of the solution. In this work, we deal with the $p$-adic scattering equation. We adapt these methods to investigate the convergence of the solutions and their qualitative properties, including mass conservation, regularity ans stability. Most of these results follow from the General Relative Entropy Inequality. We also show the existence of Feller process attached to the $p$-adic scattering equations