The Aronson–Bénilan estimate in Lebesgue spaces

In a celebrated three-pages long paper in 1979, Aronson and Bénilan obtained a remarkable estimate on second order derivatives for the solution of the porous media equation. Since its publication, the theory of porous medium flow has expanded relentlessly with applications including thermodynamics, gas flow, ground water flow as well as ecological population dynamics. The purpose of this paper is to clarify the use of recent extensions of the Aronson and Bénilan estimate in $L ^p$ spaces, of some modifications and improvements, as well as to show certain limitations of their strategy.