  In the present work we establish sharp regularity estimates for the solutions
of the porous medium equation, along their zero level-sets. We work under a
proximity regime on the exponent governing the nonlinearity of the problem.
Then, we prove that solutions are locally of class
$\mathcal{C}^{1-,\frac{1}{2}-}$ along free boundary points $x_0\in
\partial\left\lbrace u>0\right\rbrace$, both in time and space. Our argument
consists of importing information from the heat equation, through approximation
and localization methods.
